WASHINGTON — With Democrats moving away from an insistence on a timetable for withdrawing American troops from Iraq, the Senate majority leader, Harry Reid, hit back at one of the contenders for his party’s presidential nomination, John Edwards, who has urged Congress to send President Bush the same bill over and over until he signs it.

“I care a great deal about John Edwards, as well as his wife and his children. He’s not in the Senate. I am,” Mr. Reid told reporters at a briefing in his Senate office yesterday.

Mr. Edwards, a former North Carolina senator who left office in 2005, has released a campaign ad calling on Democrats to send Mr. Bush the bill he vetoed earlier this week “again and again.”

That legislation made funding for the war conditional on a timeline for troop withdrawal, which the White House has adamantly opposed. A veto override vote in the House failed on Wednesday, and Democrats are not expected to include the same withdrawal requirement in a new bill.

“He doesn’t have to cast votes here in the Senate. We do,” Mr. Reid said of Mr. Edwards.

In a statement later in the day, Mr. Edwards stood by his position. “I am very concerned about reports that a timetable for withdrawal is off the table — ending the war should be non-negotiable,” he said.

Despite his comments about his former colleague, Mr. Reid said he had “no concern” about the four Democratic presidential candidates still serving in the Senate: Hillary Clinton, Barack Obama, Christopher Dodd, and Joseph Biden.

“They’ve caused me no concern,” he said. “They’ve caused the caucus no concern.”

The presidential posturing on Iraq, he said, has remained on the campaign trail.

“They can kick and scream and bite and scratch out there politically,” the Nevada lawmaker said. “But they haven’t done that within the caucus.”

His comments came hours before Mrs. Clinton took to the Senate floor to announce that she was co-sponsoring legislation with Senator Byrd of West Virginia that would deauthorize the war by October. The bill would put a five-year expiration date on the authorization for military force that Congress passed in October 2002.

“I believe this fall is the time to review the Iraq war authorization and to have a full national debate so the people can be heard,” Mrs. Clinton said in a floor statement.

A rival Democrat, Mr. Dodd, said he supported Mrs. Clinton’s effort but that “sadly, it will not change the president’s course in Iraq.” The Connecticut senator is backing a bill by Mr. Reid and Senator Feingold that would require Mr. Bush to redeploy troops within a year. Neither Mrs. Clinton nor Mr. Obama have signed on to that proposal.

The jousting followed the first substantive negotiating session yesterday morning between the White House and Senate leaders on a new emergency funding bill. Mr. Reid described a 45-minute meeting with presidential aides as “constructive” and “comfortable,” but they reached no agreement.

While he hesitated to discuss specifics, he resisted the suggestion that Democrats had ruled out trying to keep a withdrawal timeline in the bill. “There is nothing that’s off the table, including timetables. Nothing,” Mr. Reid said.

Yet the focus had clearly shifted to other areas, particularly the inclusion of “benchmarks” for the Iraqi government that could draw bipartisan support.

The party leadership is also likely to cut nonmilitary spending that Republicans and the White House have derided as pork, a senior Democrat told The New York Sun. Congressional leaders may also strip the binding timetables from the bill, instead pushing for them to be included in a defense spending authorization that lawmakers will take up this summer, the Democrat said. Such a move will allow extra time to observe developments in Iraq and pressure more Republicans to support a date for withdrawal.