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Rep. Darrell Issa said Friday he's considering jumping into the race for House speaker, following the surprise withdrawal by fellow California Republican, House Majority Leader Kevin McCarthy.

"To be honest, I'm probably better suited to be the Republican leader or a committee chairman," said Issa, whose four-years as House Oversight chairman ended in late 2014.

Issa, an eight-term congressman, told CNBC's "Squawk Box" he would support House Ways and Means Chairman Paul Ryan for speaker, though the Wisconsin Republican said Thursday after McCarthy bowed out that he did not want the job.

"A lot of people are still urging" Ryan, the 2012 vice presidential nominee in his ninth term in Congress, to reconsider, Issa said.

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Four-term Rep. Jason Chaffetz of Utah, current House Oversight chairman, and three-term Rep. Daniel Webster of Florida were running against McCarthy, who was first elected to Congress in 2006.

Issa said he did not have a second choice among those announced candidates — stressing the Republican Party needs to find a consensus candidate to bring the staunch conservative wing of the party back into the fold.

If that single candidate can not be found, Issa said he'd advocate a complete leadership overhaul. McCarthy has said he plans to stay on as majority leader. "Nobody, in my opinion, covets the job so much that they would throw our future under the bus," Issa added.

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"It's an important time for Republicans. We have a mandate that we haven't yet met after the last two elections," said Issa, referring to the GOP gaining control of the House the 2010 midterm elections and the Senate last year. "It is about not compromising on your values but compromising on how much everybody gets in a process of moving in the right direction."

Rep. Peter King from New York echoed those sentiments. Republican and Democrats are supposed to work together, he told CNBC on Friday. "The idea is get the best you can and make deals. They're nothing wrong with making deals. That's what politics is. It's compromise. That's what democracy is."

Outgoing House Speaker John Boehner, whose bombshell resignation announcement last month set off the race, said he'd stay on until a successor is elected. The 13-term lawmaker from Ohio had planned to leave Congress at the end of October.

"I would say this is unprecedented," said King, who's serving in his 12th term. "In the past you did have fights within the party over [leadership] ... but once the decision was made within the party then they went to the House floor united."

He added: "Boehner had support of 80-85 percent of Republicans, yet he was forced to step down because one faction was basically willing to bring the House down, shut the government down." He said McCarthy had enjoyed "easily 85 percent of votes" but was throttled by "30-35 people who said when it came to the House floor they would not vote for him."

The winner needs 218 votes.

The House Freedom Caucus, highly critical of Boehner, opposed McCarthy and supported Webster for speaker.

Issa said the caucus members are not the problem. "They are people who, in fact, simply want to be heard. ... [They] are necessary to be a constructive part of a Republican conservative agenda."

Issa blamed Boehner for alienating conservatives by using "terms for unnamed members that were somewhat pejorative" when the speaker felt his agenda was being stymied.

"He began to forget that it's a team. The entire leadership team has to be recognized at the table," said Issa. He said McCarthy and House Majority Whip Steve Scalise of Louisiana, in Congress since winning a 2008 special election, told Boehner he needed to do things differently.

But Issa said no one questions Boehner's motives or his commitment to the institution.

In addition to duties leading the House, the speaker is second in line for the president, after the vice president.