WASHINGTON — On this, the two presidential rivals, Senators Obama and Clinton, can agree: They sometimes disagree with their trash-talking supporters and will try to cool it.

Advisers to the Democratic candidates shed some light today on the private chat the two candidates had yesterday on the Senate floor. The talk lasted three or four minutes in full view of reporters watching on the balcony above who could see them talking, but not hear what they said.

“They approached one another and spoke about how supporters for both campaigns have said things they reject,” a Clinton spokesman, Phil Singer, said. “They agreed that the contrasts between their respective records, qualifications and issues should be what drives this campaign, and nothing else.”

An Obama adviser, speaking on a condition of anonymity about the private conversation, gave a similar account, while stressing that it was Mr. Obama who approached Mrs. Clinton on the subject. They committed to making sure that their supporters don’t get overheated in the future, the adviser said.

With Mrs. Clinton and Mr. Obama in such a close race for the nomination, the campaign has grown increasingly acrimonious. Surrogates and aides to the candidates have stirred racial and gender divisions that Democrats fear will leave the eventual nominee badly damaged and battered heading into the general election.

Party members also fear that the rancor will make it difficult to unite the party for the November contest, undercutting Democrats’ chances of winning the White House after eight years of President Bush.

Democrats face a protracted nomination fight that is likely to last at least until voting ends in June and perhaps won’t be resolved until the convention in Denver at the end of August.

In contrast, Senator McCain has already secured the Republican nomination and has moved forward with fundraising and other efforts that will prepare him to campaign against whoever emerges from the Obama-Clinton fight. A badly damaged and battered nominee will do nothing to help their united cause of getting a Democrat back in the White House next year.

Mrs. Clinton and Mr. Obama’s rare appearance in the Senate was to cast several votes on a closely debated budget bill.

It followed a week of harsh words that led to resignations in both camps. Samantha Power gave up her position as an unpaid foreign policy adviser to Mr. Obama after calling Mr. Clinton a “monster.” Geraldine Ferraro, who was the party’s vice presidential nominee in 1984, stepped down from Mrs. Clinton’s finance committee after she said Mr. Obama would not have made it this far if he were white.