Washington (CNN) Ben Carson said Monday he could indeed support a Muslim for president -- despite words to the contrary on Sunday -- should they pledge fealty to the Constitution.

Carson, under fire for seeming to suggest that he couldn't support anyone who subscribes to Islam to lead the White House, issued a caveat to that position on Monday evening. In an interview with Fox News' Sean Hannity, the Republican presidential candidate said there are some Muslims who could be president if they effectively renounced their faith.

"If someone has a Muslim background and they're willing to reject those tenets and to accept the way of life that we have and clearly will swear to place our Constitution above their religion ... I would then be quite willing to support them," he said.

"I could never support a candidate for President of the United States that was Muslim and had not renounced the central tenant of Islam: Sharia Law," he wrote. "I know that there are many peaceful Muslims who do not adhere to these beliefs. But until these tenants are fully renounced...I cannot advocate any Muslim candidate for President.

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