Senate leaders slowly stepped away from a major showdown over presidential nominees on Tuesday, as Republicans dropped their opposition to a key appointee and Democrats offered a significant concession in return -- the withdrawal of two controversial labor board picks.

By Tuesday afternoon, lawmakers were describing the arrangement as a tentative "deal" that could avert an ugly stand-off. Senate Majority Leader Harry Reid earlier had demanded that Republicans advance and approve at least seven nominees or face a controversial rule change that could drain their power. But Reid confirmed late Tuesday that things were cooling down.

"We have a new start for this body, and I feel very comfortable with it," Reid said.

On the Republican end, several members of the GOP joined Democrats to advance, in a 71-29 vote, the nomination of Richard Cordray, Obama's pick for director of the Consumer Financial Protection Bureau. He was later approved in a 66-34 vote.

In return, President Obama to withdrew two controversial nominees to the National Labor Relations Board, Richard Griffin and Sharon Block. In their places, Obama nominated Nancy Schiffer, a former top lawyer for the AFL-CIO, and Kent Hirozawa, NLRB counsel. Senate Republican Leader Mitch McConnell said Tuesday he was pleased by this apparent concession.

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Aides have said the supposed "deal" is still being finalized. Sen. Lindsey Graham, R-S.C., said earlier he thinks the agreement is "fragile."

At issue is the Senate's own rulebook. If Republicans don't confirm the nominees, Reid was threatening to flout the normal process in order to change the rules so key presidential nominees can be confirmed with just 51 votes -- as opposed to 60. In short, the move would curb the minority party's ability to filibuster nominees.

Critics say this would likely prompt Republicans to retaliate by doing even more to reduce the minority party's rights when the GOP regains control of the Senate. That could happen as early 18 months from now, after the 2014 elections.

"It's a decision that, if they actually go through with it, they will live to regret," McConnell said of Democrats.

Reid signaled Tuesday that he did not plan to go through with the threat any time soon. But he also signaled he's willing to renew the threat if necessary.

"(Republicans are) not sacrificing their right to filibuster and we are damn sure not filibustering our right to change the rules if necessary," Reid said.

McConnell had warned that Reid's threatened action would "change the core of the Senate." He said it would fundamentally deny senators their right to question potential officials.

Republicans have delayed or blocked presidential nominees by using their filibuster powers, which allow 41 senators to block actions in the 100-member chamber.

Among the nominees in question were those to the National Labor Relations Board. Obama originally appointed them when he said the Senate was in recess -- but an appeals court ruled that Obama exceeded his authority, and Republicans likewise describe the appointments as illegitimate.

The roster also includes EPA Administrator nominee Gina McCarthy.

Nearly every senator gathered for an hours-long meeting on the stand-off late Monday. Earlier, White House Chief of Staff Denis McDonough also met with Reid's chief of staff. Terms of a deal did not begin to publicly materialize until Tuesday.

The proposed rule change, if Reid were to go through with it, would not end filibusters for legislation or judicial nominees. But some senators say a limited rule change now could open the gate for much deeper changes in the years ahead.

White House Press Secretary Jay Carney said Monday that Obama "supports Harry Reid and is appreciative of the support Harry Reid has given to his nominees."

Carney, asked if Obama worried that the Senate could become even more dysfunctional if rules are changed, said: "Well, it boggles the mind how they would achieve that."

The Associated Press contributed to this report.