Darrell Issa: I may run for House speaker

Rep. Darrell Issa said Friday he's considering jumping into the race for House speaker, following the surprise withdrawal by 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-year reign as House Oversight chairman ended in late 2014.

The California congressman told CNBC's " Squawk Box " he would support House Ways and Means Chairman Paul Ryan for speaker, though Ryan said Thursday after McCarthy bowed out that he did not want the job. "A lot of people are still urging [Ryan]," Issa said.

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Rep. Jason Chaffetz of Utah, current House Oversight chairman, and Rep. Daniel Webster of Florida were running against McCarthy.

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 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."

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. He had planned to resign from Congress at the end of October.

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

Sympathetic toward the conservative group, Issa said the 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 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.

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