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Ted Cruz for House speaker?

With the news that House Majority Leader Kevin McCarthy, R-California, has dropped his bid for House speaker, the Republican conference is scrambling to find a new leader.

Although speakers have always been members of the House of Representatives, it is in fact not a requirement written in the Constitution. Article I, Section 2 Clause 5 reads:

"The House of Representatives shall chuse [sic] their Speaker and other Officers; and shall have the sole Power of Impeachment."

On a few occasions, lawmakers have voted for someone who isn't a member. Last January, when John Boehner was elected to serve as speaker of the 114th Congress, various members voted for former Secretary of State Colin Powell and Sens. Rand Paul, R-Kentucky and Rep. Jeff Sessions, R-Alabama.

Sen. Ted Cruz, R-Texas, is a popular figure among the most conservative wing of the party because of his refusal to compromise. He met with about two dozen conservatives Wednesday to urge them to choose a strong, principled leader. But Cruz does have his eye on a higher office - the presidency - so he declined to be the next speaker and said he'd leave it up to the House.

This could be bad news for Boehner, who will not be able to leave at the end of October as planned if his GOP colleagues can't find a new speaker. He said in a statement Thursday that he'll stay in his job until the House elects a successor.

Who is running for speaker?

The two members who were planning to challenge McCarthy, Utah Rep. Jason Chaffetz and Florida Rep. Daniel Webster, are both continuing their bids.

"I really do believe it is time for a fresh start. That was the whole genesis of my campaign," Chaffetz said after McCarthy dropped out Thursday. "Our conference is going to have to do a lot of deep soul searching and we'll see what happens."

Who's not running?

McCarthy's decision upped the pressure on House Ways and Means Committee Paul Ryan, R-Wisconsin, who serves as a rare bridge between the leadership and more conservative members. But he was quick to reiterate he had no interest in the job.

"While I am grateful for the encouragement I've received, I will not be a candidate," he said in a statement shortly after McCarthy's announcement.

Another congressman who's attracting support is Rep. Trey Gowdy, R-South Carolina, who is the chairman of the committee investigating the Benghazi terror attacks. But he, too, says he won't run.

Who could run?

The House Freedom Caucus - the most conservative members of the House - is backing Webster, but there is still a sizable number of members who would likely prefer a speaker with less hard-line views and more leadership experience. A few members who could fit that description include Republican conference chairwoman Cathy McMorris-Rodgers, R-Washington, House Financial Service Committee Chairman Jeb Hensarling, R-Texas, Education and Welfare Committee Chairman Tom Cole, R-Oklahoma, and Rep. Peter Roskam, R-Illinois.

Rep. Kevin McCarthy on his exit from race for House speaker 02:19

Hensarling in particular has some advantages. He's from Texas, a big state with lots of representation, and he has deep ties to the conservative wing of the party. Roskam used to be chief deputy whip.

Republican Whip Steve Scalise, R-Louisiana, and House Budget Committee Chairman Tom Price, R-Georgia, who were running to take over as Majority Leader under the assumption that McCarthy would win, could also aim a step higher. Price was backed by both Ryan and Hensarling in the race for majority leader.

What about a temporary speaker?

One idea that's making the rounds in the GOP conference is the prospect of a "caretaker speaker"- someone who is planning to retire at the end of this term already and would take on the role temporarily while Republicans search for a more permanent leader. Names like Reps. Candice Miller, R-Michigan, and John Kline, R-Minnesota, have been bandied about, mostly by more moderate members who are worried about the prospect of a Webster speakership. But there would be significant downsides to this approach - the interim speaker would essentially be a lame duck on day one, giving them even less control over the unruly GOP conference than Boehner had.

Rep. Greg Walden, R-Oregon, who currently serves as the National Republican Congressional Committee Chairman, said he would consider serving as an interim speaker.

CBS News Congressional Correspondent Nancy Cordes and CBS Radio News Congressional Correspondent Steven Portnoy contributed to this story.

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