A compromise deal to confirm fewer than those seven will not suffice, Reid said only hours before a planned bipartisan gathering on the deeply divisive issue.
The Nevada Democrat again threatened to change Senate rules to allow simple-majority votes for presidential nominees other than judges. Despite virulent protests by Republicans, Reid called the move "a minor change, no big deal."
Republicans have often delayed or blocked presidential nominees by using their filibuster powers, which allow 41 senators to block actions in the 100-member chamber. Many senators cherish rules and traditions that give minority members far more power than anything enjoyed by minority-party members of the House.
A GOP agreement to confirm the seven nominees -- for posts dealing with labor relations, consumer protection and the environment -- would defuse the tension and allow the Senate to return to normal business, Reid said. He spoke at the Center for American Progress, a liberal think tank with close ties to the White House.
Senate Republicans and at least one Democrat, Sen. Carl Levin of Michigan, strongly oppose his proposed rules change. All 100 senators were invited to a Monday night meeting in the Old Senate Chamber, in a last ditch-effort to resolve the disagreement.
"The power of an extreme minority" threatens the Senate's integrity, and forces him to act, Reid said Monday. He acknowledged that future Republican presidents were likely to benefit from the proposed rule change.
"It's about making Washington work, regardless who's in the White House," he said.
The proposed change 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.
Senate Republican Leader Mitch McConnell of Kentucky said Reid's threatened action would "change the core of the Senate." He said it would fundamentally deny senators their right to question potential officials.
Reid said Republicans have abused that right in order to keep President Barack Obama from assembling his team. Reid said Republicans flooded Gina McCarthy, nominated to head the Environmental Protection Agency, with 1,100 written questions, and then complained about her responses.
"My efforts are directed at saving the Senate from becoming obsolete," Reid said Monday.
White House press secretary Jay Carney said Obama "supports Harry Reid and is appreciative of the support Harry Reid has given to his nominees."
"The president believes that the Senate ought to function and hopes the Senate will figure out a way that the nomination process is appropriately streamlined," Carney told reporters.
Carney, asked whether 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."
Reid and McConnell discussed the issue separately Sunday on NBC's "Meet the Press."
McConnell called Democrats' proposed changes contrary to important Senate traditions. "I hope that we'll come to our senses and not change the core of the Senate," he said.
"We need to start talking to each other instead of at each other," McConnell said.
In particular, Republicans have objected to a pair of union-backed members of the National Labor Relations Board, Richard Griffin and Sharon Block. Obama appointed them when he said the Senate was in recess. An appeals court has ruled that Obama exceeded his authority; the board's actions since they took their seats are in legal limbo.
Republicans also have objected to Obama's pick to lead the Consumer Financial Protection Bureau, which was created as part of Wall Street overhaul legislation that was opposed by the GOP. Obama nominated his pick, former Ohio Attorney General Richard Cordray, more than two years ago.
Many Republican senators say they will not confirm anyone to the consumer post unless the bureau's leadership structure is changed.
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