But Democrats acknowledged that they had won no concessions from Republicans to avoid a filibuster of future nominees, such as those for the Department of Homeland Security or the Social Security Administration. Reid stressed that Democrats have not ruled out another attempt at the nuclear option if he feels that is warranted.
Senate Minority Leader Mitch McConnell said the meeting Monday "led to a constructive outcome."
"I thought it was really good for the institution for us to be talking to each other rather than at each other," he said.
But he also noted several times: "None of our rights will be waived."
Democrats have said Perez has a history of bipartisanship and is well qualified for the labor job. Republicans have countered that the Civil Rights Division he leads has made a series of partisan decisions, and that Perez has not been entirely forthcoming about his role in those decisions.
Sen. Tom Harkin, the Iowa Democrat who chairs the Senate committee that advanced Perez's nomination on a party-line vote in May, said Tuesday he was pleased with the filibuster agreement in part because it would allow his nomination to move forward.
It "is very important because [Republicans] were coming after him," Harkin said. "They were out to stop him."
Republicans have raised several concerns about Perez, but have focused on a deal by the Justice Department to back out of a lawsuit filed against the city of St. Paul, Minn., in exchange for city leaders' dropping separate litigation against the federal government.
Republicans have said taxpayers could have recovered $200 million in misspent funds if the Justice Department had pursued the case. Perez has said he was concerned the city's lawsuit could have resulted in an adverse Supreme Court ruling. He says he cleared the move with ethics officials in the department.
McConnell used a floor speech this year to call Perez a "crusading ideologue."
Perez, a native of Buffalo, N.Y., was the first Hispanic to win a seat on the Montgomery County Council in 2002 and ran briefly for Maryland attorney general in 2006. His grandfather was a Dominican ambassador to the United States.
He would replace former Labor Secretary Hilda L. Solis, who resigned in January.