Gov. Gretchen Whitmer of Michigan stood glumly in the Oval Office, hoping no one would take her picture.

She had not expected to be there, standing in front of the cameras, as President Trump signed executive orders punishing those who opposed his 2020 election lies. Ms. Whitmer, a prominent Democrat seen as a possible 2028 presidential candidate, had come to the White House to discuss funding for an Air National Guard base near Detroit and aid for thousands of Michiganders who had just been hit by an ice storm.

Then Mr. Trump’s aides surprised her on Wednesday by ushering her into the Oval Office not for her scheduled one-on-one meeting with the president, but for a politically loaded appearance before the press corps. She found herself an unwilling participant in his unending reality show, with photos of her rocketing around group chats of Democratic strategists who wondered what on earth she was doing.

The episode was the result of a remarkable attempt at reconciliation between Ms. Whitmer and Mr. Trump, who dismissed her in 2020 as “that woman from Michigan” during a clash over his administration’s pandemic response.

The day after the inauguration, Ms. Whitmer penned a handwritten letter — which has not been previously reported — congratulating Mr. Trump, saying she looked forward to working together and praising his support for the auto industry in his first address, according to a person who relayed the text of the letter. Ms. Whitmer included her cellphone number and invited Mr. Trump to call her if she could be of any help to him.

The outreach worked for her, but it came at a cost.

Her whipsaw experience with Mr. Trump illustrates the political risks that Democratic governors face as a small group of them try to cultivate relationships with a president reviled by their party but in control of vast amounts of federal funding for states.

These governors — exemplified by Ms. Whitmer but also including Gavin Newsom of California, Phil Murphy of New Jersey and Kathy Hochul of New York — have met with the president in the Oval Office, fielded his phone calls and toned down their language toward him.

Many Democrats see Mr. Trump as a transactional politician who is susceptible to flattery and are acutely aware of how he has sought to punish liberal states and groups. He has aimed to cut off billions of dollars to universities in states with Democratic governors and threatened funding for local public education and public health, leaving state leaders scrambling to find alternative sources of cash or cut spending in other areas.

But Democratic governors also have their own political ambitions to consider. Many in the party see their state leaders as the best hope to win back the White House in 2028, and the liberal base increasingly wants elected officials to aggressively fight Mr. Trump’s actions.

That is why a second group of Democratic governors, led by JB Pritzker of Illinois, Josh Shapiro of Pennsylvania and Tim Walz of Minnesota, has been more publicly antagonistic toward Mr. Trump. Their tougher stance may reflect how they have one eye on their own re-election bids in 2026 and another on the 2028 Democratic presidential primary contest.

“Democrats from the center to the left believe Trump is an autocrat who represents an existential threat to democracy and our rights,” said Neera Tanden, the chief executive of the Center for American Progress, a top liberal think tank. “They expect their leaders to meet the moment by fighting his dictatorial attacks, not placate, negotiate or assuage because doing so makes him stronger and bully others more.”

In an interview on Friday, Ms. Whitmer said she had no regrets.

“Public service is about putting the people of Michigan before my own interest,” she said. “My job was to try to get help for people who were suffering as a result of the ice storm, to land more investment at Selfridge air base, to protect the Great Lakes and to fight for the auto industry. And that’s what I was doing.”

She added, “I’m always going to show up for the people of Michigan, and that’s probably why I got elected by double digits.”

Former President Joseph R. Biden Jr. dealt with Republican governors much differently, barely speaking with them except about disaster relief.

But with Mr. Trump in office, Democratic governors have found more opportunities.

Several beyond Ms. Whitmer, including Ms. Hochul, Mr. Murphy and Mr. Newsom, have had one-on-one meetings with Mr. Trump in the Oval Office, with one of Ms. Hochul’s sessions lasting two hours.

The president also calls Democratic governors, and picks up calls from them, with some frequency. He has given several of them his cellphone number.

Mr. Murphy has a longstanding relationship with Mr. Trump, forged during the pandemic and bolstered by the president’s frequent trips to his golf club in New Jersey. After the president was shot at a campaign rally in Butler, Pa., in July, Mr. Murphy and his wife, Tammy, visited him in Bedminster, N.J.

With Mr. Trump back in office, Mr. Murphy has sought to find common ground with him in opposing congestion pricing in New York City and fixing sinkholes on Interstate 80 in New Jersey.

When Mr. Murphy and his wife were at the White House in February for a National Governors Association dinner, the New Jersey governor invited Mr. Trump to an upcoming ribbon-cutting for the new Portal North Bridge over the Hackensack River. Ms. Murphy also invited Mr. Trump to come to an Ultimate Fighting Championship event in June in Newark.

Mr. Trump expressed interest — but the governor’s outreach has not spared him the Trump administration’s ire. The top federal prosecutor in New Jersey, a Trump ally, said on Friday that she planned to investigate Mr. Murphy over immigration policy.

In December, before Mr. Trump was inaugurated, Mr. Newsom talked himself up as a leading figure in the Democratic opposition to Mr. Trump. But his posture changed after fires ravaged greater Los Angeles in January, leaving the California governor in need of nearly $40 billion in federal aid.

When Mr. Trump and his wife, Melania, arrived to review damage from the blazes, Mr. Newsom met them at the airport and kissed Mrs. Trump on the cheek. Shortly after, he traveled to Washington for a 90-minute meeting in the Oval Office. Mr. Newsom has also hosted Trump allies including Charlie Kirk and Stephen K. Bannon on his new podcast.

Ms. Hochul has had two Oval Office meetings with Mr. Trump and is in regular contact with him. She said in an interview that he had called her occasionally to check up on New York projects that interested him. Last week, on the day he announced far-reaching tariffs, she recalled, Mr. Trump phoned her to ask questions about Penn Station and Amtrak.

Asked to describe their lengthy conversations, Ms. Hochul replied, “I listen a lot.”

She went on: “I’m always willing to engage and talk and listen and find out areas where I can do what I have to do, which is, No. 1, to protect New York State. When you challenge our values or our policies, then we have a fight. But I can also be adversarial without being acrimonious, and work toward common areas.”

Steven Cheung, the White House communications director, said in a statement: “It’s no surprise every elected official in America — Republican or Democrat — wants a productive working relationship with President Trump. Because President Trump is the president of all Americans, he continues to work with anyone that is willing to help advance policies that benefit the American people.”

Tensions and an ‘Emergency Cocktail Hour’

Plenty of other Democratic governors have kept their distance from Mr. Trump.

Aides to Gov. Maura Healey of Massachusetts, Gov. Michelle Lujan Grisham of New Mexico, Mr. Pritzker, Mr. Shapiro and Mr. Walz, among others, said each politician had yet to have a one-on-one meeting or phone call with Mr. Trump since he returned to power.

Mr. Shapiro, Mr. Pritzker and Mr. Walz, who are seen as some of the party’s most ambitious leaders, have been among the more forceful voices pushing back against Mr. Trump.

And despite Mr. Trump’s more conciliatory approach with some Democratic governors, he has still lashed out at others, especially when it serves his political agenda.

In February, he hosted a bipartisan group of governors at the White House as part of a weekend of events coordinated by the National Governors Association. During the meeting, Mr. Trump attacked Gov. Janet Mills of Maine, a Democrat, over her state’s policies on transgender athletes’ participation in sports.

After the dust-up, Ms. Healey convened what three people with knowledge of the planning called “an emergency cocktail hour” to discuss how to respond. Ms. Healey urged fellow governors to skip a dinner at the White House that night in protest.

“It was appalling and shocking what happened to Governor Mills and the way she was treated during that business meeting,” Ms. Healey said in an interview on Friday. “I think there was a feeling coming out of it shared by a lot of us that was just really wrong and we needed to come together and process some of that.”

But many Democrats attended anyway — and after the dinner, Mr. Trump led a bipartisan group of governors and their spouses on a nearly hourlong, impromptu tour of the White House. The president took them through the residence, showing off the Lincoln Bedroom and the Yellow Oval Room while regaling the group with historical facts, and finished the jaunt in the Oval Office.