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Hillary Clinton attended a campaign event in Pittsburgh on Tuesday. Credit Eric Thayer for The New York Times

Senator Bernie Sanders, center, after meeting with Hillary Clinton in Washington on Tuesday. Credit Zach Gibson/The New York Times 

WASHINGTON — With little affection or trust between them, Hillary Clinton and Senator Bernie Sanders met privately on Tuesday night to size each other up as they started exploring what kind of alliance they might build for the general election battle against Donald J. Trump.

Neither Democrat entered the meeting on sure footing, and both were a little tense, advisers in each camp said beforehand.

Mrs. Clinton, who became the presumptive Democratic presidential nominee last week, wanted to know what it would take to earn Mr. Sanders’s endorsement and whether he would seek policy concessions or political promises, her advisers said. Mr. Sanders wanted to gauge the depth of Mrs. Clinton’s commitment to progressive goals like a higher minimum wage and lower financial burdens on college students, and to making the Democratic nomination process more open in the future.

The chemistry between the two candidates was strained, in part, because Mrs. Clinton and Mr. Sanders had not had any real chances to air grievances or blow off steam with each other away from the television cameras during their 14-month fight for the nomination.

Mrs. Clinton had a few such moments with Barack Obama in 2008 before they sat down for their own post-nomination tête-à-tête, which made it a little easier for them to come together, unite the Democratic Party and win that November.

In a sign that they are still adjusting to each other, Mrs. Clinton and Mr. Sanders were joined in their meeting by Jane Sanders, Mr. Sanders’s wife; Jeff Weaver, his campaign manager; John D. Podesta, Mrs. Clinton’s campaign chairman; and Robby Mook, her campaign manager.

Two advisers to Mr. Sanders described him as concerned that Mrs. Clinton might say all the right things now but embrace more politically moderate positions later if she thinks it necessary to win states like Florida, Ohio and Virginia.

The advisers, who spoke on the condition of anonymity because the campaign had not authorized them to speak, said Mr. Sanders felt no pressure to endorse Mrs. Clinton quickly. He wants her to take steps to win his confidence in the five and a half weeks before the Democratic convention, where his voters and delegates expect him to speak and Clinton advisers hope he will give a full-throated speech backing her.

Mr. Sanders has leverage: He accrued about 12 million votes and 1,877 delegates, and in a New York Times/CBS News poll last month, 28 percent of his supporters said they would not vote for Mrs. Clinton if she was the Democratic nominee. Mrs. Clinton picked up nearly 16 million votes and 2,784 delegates.

Whether Mr. Sanders endorses her enthusiastically and campaigns for her, or recognizes her as the nominee but otherwise withholds his blessing, is a significant concern for some Clinton advisers. Others in her campaign think that Democrats will ultimately unite because the possibility of a Trump victory is too great to ignore.

Setting the stage for their meeting, Mr. Sanders used a news conference here Tuesday afternoon to call for replacing the leaders of the Democratic National Committee, eliminating the role of superdelegates in the party’s nomination process, and allowing independents and last-minute registrants to vote in all Democratic primaries. His campaign has long viewed the head of the committee, Representative Debbie Wasserman Schultz of Florida, as a Clinton ally who orchestrated the debate schedule and made other decisions to benefit Mrs. Clinton.

“I think the time is now — in fact, the time is long overdue — for a fundamental transformation of the Democratic Party,” Mr. Sanders said at the news conference. “We need a party which is prepared to stand up for the disappearing middle class, for the 47 million people in this country who are living in poverty, and take on the greed of the powerful special interests that are doing so much harm to this country.”

Mrs. Clinton and Mr. Sanders met on the day of the Democratic primary in Washington, D.C., the final contest of the four-and-a-half-month nominating process. Shortly after voting ended, The Associated Press declared Mrs. Clinton the winner; with three-quarters of precincts reporting, she had won 79 percent of the vote.

Mr. Sanders received a standing ovation when he dropped in at the Senate Democrats’ weekly lunch to speak about his campaign and pose for an official Senate photo with his colleagues. He has not been in the Capitol much of late; according to his website, he has not cast a vote since Jan. 12.

“It was very upbeat, very optimistic,” Senator Gary Peters, Democrat of Michigan, said.