At the top of some Republicans' wish list is former Florida Governor Jeb Bush, who is now championing education reform.
After Indiana Governor Mitch Daniels announced his decision Sunday not to run, Bush reiterated his intentions.
"While I am flattered by everyone's encouragement, my decision has not changed. I will not be a candidate for president in 2012," Bush said in a statement.
Republican Rep. Paul Ryan, chairman of the House Budget committee, has also received attention since he gained more national recognition for championing a proposed federal budget that would slash the federal debt by cutting spending and reforming Medicare and Medicaid. While his blueprint has been the focus of Democratic attacks, his long championing of the fiscal issue has garnered him praise from many Republicans.