New speaker race: Webster, Chaffetz & Ryan?

House Majority Leader Kevin McCarthy threw the race for the next speaker of the House into disarray by withdrawing his bid for the position on Thursday. Two of his colleagues officially remain in the race while political analysts speculate about others who might throw their hat into the ring.

Read MoreGOP's McCarthy dropping out of race for House speaker

After McCarthy drops out, where does the GOP turn?
After McCarthy drops out, where does the GOP turn?

Paul Ryan of Wisconsin

Paul Ryan
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In a statement, Paul Ryan stressed that he would not pursue the position of speaker of the House, calling it a hard "no." Ryan said he was "disappointed" in McCarthy's decision, calling him "the best person to the lead the House."

But some say if Ryan faces enough pressure from Republicans, he might throw his hat into the ring. According to a Washington Post report which NBC News confirmed, current Speaker Boehner has personally asked Ryan to run for the position. Furthermore, Ryan has canceled all his fundraisers for the next 48 hours.

Brian Gardner, Washington Research Analyst at KBW, told CNBC's "Power Lunch" on Thursday that, "There was no obvious alternative to Kevin McCarthy. I think Paul Ryan's name is going to come back into it."

Gardner added, "I know Congressman Ryan has said 'no' in the past. I know there's going to be a lot of pressure brought to bear on him."

Daniel Webster of Florida

Rep. Daniel Webster, R-Fla. speaks to a reporter as he leaves a House Republican special leadership election meeting on Capitol Hill in Washington, Thursday, Oct. 8, 2015.
Andrew Harnik | AP

Daniel Webster announced his bid for the post Monday.

"As speaker of the House, I would commit to serving you by promoting principle over power," he said in a video published on his Twitter account.

Webster Tweet

Webster said on his website that he aims to "rein in out-of-control bureaucratic regulations that stifle the ability of small businesses to grow our economy and create jobs."

He added that another one of his goals is to encourage small businesses to hire by providing tax relief. This includes "[simplifying] and [flattening] our tax system through comprehensive tax reform."

Webster's other policy ideas include safeguarding financial security for seniors, strengthening the dollar, and recovering foreign profits and return on investment as well as bringing jobs back to America.

"In order to truly revitalize our economy for long-term sustainability, we must remove roadblocks that business and entrepreneurs face," Webster wrote.

Jason Chaffetz of Utah

Jason Chaffetz
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Jason Chaffetz announced his candidacy Sunday in a statement posted on his website.

Chaffetz said he is running for this position in order to tackle "the toughest issues facing the United States of America."

"I came to Congress to help fix problems, and as speaker I will fight every day to make that happen," he added.

He also said that the federal government has a spending problem with the biggest challenge being entitlement expenditures. In November 2011, Chaffetz announced a seven-point Social Security reform proposal that "avoids tax increases and trust fund insolvency."

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"There is no excuse for allowing Social Security to become insolvent," Chaffetz said. "The program is unsustainable in its current form. However, this problem is completely within our power to resolve — without tax increases and while protecting retirees. This series of simple steps will make the program solvent and allow younger generations to more accurately anticipate their own retirement needs."

Trey Gowdy of South Carolina

Trey Gowdy
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Another name being thrown around for the race is Trey Gowdy.He wrote on his website that to spur job creation, Congress must reform the tax code, "which stifles job creation and includes one of the highest corporate tax rates in the world.

"Congress can support job creation by reducing unnecessary government regulations, expanding American energy production and encouraging innovation. Congress must also work to rein in out-of-control spending, which has fed an [$18 trillion] national debt and threatened the long-term stability of our economy."

Tom Cole of Oklahoma

Tom Cole
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Tom Cole, too, believes that Americans are taxed "too much."

"I have supported legislation that would drastically simplify the tax code, consolidate the six tax brackets into two, repeal the alternative minimum tax and support economic growth while not raising taxes. By closing loopholes and lowering both individual and corporate tax rates, we can incentivize America's job creators to do just that," he wrote on his website.

Cole said small businesses should get help to provide their employees with health care.

"I will continue to support policies that would help small businesses provide health care to their employees through health savings accounts, association health plans or other tax deductions for medical expenses," he said.

Read MoreHow divided is the GOP field on Medicaid?

Cole would also like to see "unfair foreign trade practices" done away with in order to promote American exports.

"I support trade policies that would increase our exports, especially those that affect Oklahoma, like agriculture and energy technology. Generally speaking, I will be supportive of future free trade agreements that strengthen our domestic economy and increase the ability of American citizens to move products into overseas markets."

Peter Roskam of Illinois

Peter Roskam
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Peter Roskam has said "our burdensome tax code is holding our economy back." That most Americans need a computer program or tax professional to do their taxes is proof of how complicated the system is, he wrote on his website.

"Studies have shown that reforming both our corporate and individual tax code has the potential to unleash economic growth and job creation," he wrote. "By closing loopholes and lowering rates, we can create a tax policy that is simpler, flatter and fairer for everyone."

Roskam supports President Barack Obama's goal to double exports by 2015. He backed the Trade Promotion Authority (TPA) legislation introduced last year.

"Armed with TPA, the United States can build upon recent trade success like the 2012 passage of the Colombia, Panama and South Korea free trade agreements, opening up new markets for American products, and boosting U.S. exports, all of which requires businesses to expand and hire to meet growing demand," he said.

Jeb Hensarling of Texas

Jeb Hensarling
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Jeb Hensarling said that the economy needs a different strategy than that promoted by the Obama administration.

"Our economy will not fully recover and robust job creation will not occur until Washington stops threatening and punishing job creators with laws and regulations that discourage economic growth," he wrote on his website.

He supported a bill that would allow Congress to vote on proposed regulations having an economic impact of $100 million or more annually before they could take effect.

Hensarling is opposed to raising taxes. "I strongly support efforts to reform our nation's tax code through pro-growth tax reform that will lower individual and corporate tax rates while eliminating the confusing and burdensome tangle of loopholes, resulting in a tax code that is fairer, flatter and simpler for families and small businesses," Hensarling said.

Read More How tax reform could shape the GOP field

He has also served on committees to reduce the deficit. "I have been a leader in the fight against Washington's spending spree in order to save the American dream as it drowns in a sea of red ink. I am fighting to protect the family budget from the federal budget to ensure we do not allow the torch of liberty to be mortgaged as a result of reckless Washington spending and so the American dream will be bigger and brighter for the next generation."

— CNBC's Jacob Pramuk contributed to this story.