Updated at 5:51 p.m. ET
Facing a backlog of presidential nominations, Senate Majority Leader Harry Reid, D-Nev., is seriously considering changing the rules to the Senate to get the nominations past Republican opposition.
"My efforts are directed at saving the Senate from becoming obsolete - a minor change. No big deal," Reid said Monday.
Others, however, consider it a really big deal. Reid's Republican counterpart, Sen. Mitch McConnell, R-Ky., suggested the move could effectively destroy the Senate.
"No majority leader wants written on his tombstone that he presided over the end of the Senate," McConnell. "Well, if this majority leader caves to the fringes and let's this happen, I'm afraid that's exactly what they'll write... I just hope the majority leader thinks about his legacy, the future of his party, and - most importantly - the future of our country before he acts."
Using the so-called "nuclear option" to change the rules -- a procedural maneuver that would force a rules change with a simple 51-vote majority -- is a big enough deal that the Senate is convening a rare bipartisan meeting in the Old Senate Chamber Monday evening to talk it over.
The last bipartisan meeting of the full Senate in the Old Senate Chamber was in 2007 for a meet-and-greet at the start of the 110th Congress. Before that, the last bipartisan meeting in the chamber was on January 8, 1999, for both parties to discuss the framework of the Clinton impeachment trial.
At an event hosted by the liberal think tank the Center for American Progress, Reid on Monday suggested he'll present Republicans at the meeting with a take-it-or-leave-it scenario: "If the sky's falling and they think it's falling, let them stop the filibusters."
Reid is specifically calling on Republicans to drop opposition to seven of President Obama's cabinet nominees: Richard Cordray, to lead the Bureau of Consumer Financial Protection; Richard Griffin, Jr., to serve on the National Labor Relations Board (NRLB); Sharon Block, to serve on the NRLB; Mark Pearce, to serve on the NRLB; Regina McCarthy, to serve as EPA Administrator; Thomas Perez, to serve as Secretary of the Department of Labor; and Fred Hochberg, to serve as president of the Export-Import Bank.
The Senate leader has scheduled showdown procedural votes for Tuesday on all seven nominations. He said Monday that if Republicans don't let all seven proceed, he'll invoke the "nuclear option" and change the rules so that it would take just 51 votes to approve an executive branch nominee.
"The status quo won't work," Reid said. Before President Obama's tenure, 20 executive nominees had been filibustered, he said; by comparison, 16 of Mr. Obama's nominees have faced a filibuster. "What they have done is just really unbelievable," he said of Republicans.
This rule change would only affect executive branch nominees -- not judicial nominees or legislation.
"That's now where the big plug is," Reid said of executive branch nominations. Potentially changing the filibuster rules for legislation, he added, is "something we could deal with later."
Later Monday afternoon, Sen. John McCain, R-Ariz., met separately with Reid and then with McConnell in an attempt to find a compromise over the nominations issue. McCain said there's a group of about 10 members, made up of Republicans and senior Democrats, interested in forging some type of deal to avert a showdown on the rules change.
"A lot of the more senior Democrats are very interested in preventing this," he said. "I always have to be optimistic, but I will agree that it's harder than last time."
Noting that 60 percent of senators were elected in the last six-and-a-half years, McCain said, "There's not the kind of institutional memory, and I do see a break between the newer members who have been here for a short period of time and the those who have been here for a longer period of time. I see a real difference there. Part of it they're new, part of it's the prevailing environment, and part of it is they've never been in the minority."
McCain said he's also talking to White House chief of staff Dennis McDonough.
CBS News producer John Nolen contributed to this report.