Hagel, who is still considering his options for the 2008 race, left open the possibility of becoming an independent and sharing a ticket with New York City Mayor Michael Bloomberg.
"I am not happy with the Republican Party today," Hagel said. "It's been hijacked by a group of single-minded almost isolationists, insulationists, power-projectors."
Hagel said he will not decide if he is going to run for president until late this summer. He did not say if he would run as a Republican or as an independent. Asked by Bob Schieffer if independent candidates would be good for the political system, Hagel said they would be.
"What America will be looking at and wanting and demanding is honest, competent, accountable leadership," he said. "We need some new, fresh, independent ideas to lead this country forward."
After dining with former New York's mayor, who is also said to be considering a run for president as an independent, Hagel said people might want to consider the two on a ticket.
"We didn't make any deals, but I think Mayor Bloomberg is the kind of individual who should seriously think about this," Hagel said. "He is the mayor of one of the greatest cities on earth. He makes that city work. That's what America wants."
He said, "It's a great country to think about - a New York boy and a Nebraska boy to be teamed up leading this nation."
The Washington insiders on the Face The Nation roundtable said they could see the attraction of a third-party ticket.
"It's the kind of thing that it's so tantalizing, I think it meets a need not just with the political system but with the public," said Michelle McQueen Martin of National Public Radio. "I mean, you see a yearning within the public for another option."
But prospects for a third-party victory don't appear to be strong.
"Third-party candidates tend not to win," Martin said. "What they do tend to do is change the dynamic of a race to favor somebody who's already in the race."
Congress's handling of the Iraq war funding bill appears to be the breaking point in the Republican party.
"For the senators, it's particularly important, because they have to rely upon independent voters," said Jeanne Cummings of politico.com. "And independent voters in New Hampshire and in Minnesota and some of these places where there're going to be very tough races, they have shifted. They're anti-war."
Hagel, who has already broken ranks with the president over Iraq, said the White House is running out of time with its war policies.
"The president may find himself standing alone sometime this fall where Republicans will start to move away, and you're starting to see trapdoors and exit signs already with a number of Republicans," Hagel said. "The 11 House Republicans that went to see him speak for more than just 11 House Republicans. That's just the tip of the iceberg."