The controversial measure also requires the doctors to have hospital admitting privileges, which could create a problem for the Jackson Women's Health Organization, which only has one physician with privileges.
CNN said the clinic owner Diane Derzis vowed to continue to provide services for women who would otherwise have to leave the state to seek abortions.
Bryant said in a written statement the bill was aimed at ensuring patient safety should a complication arise and the woman require hospitalization.
CNN noted Bryant has said in the past he wanted to make Mississippi "abortion free."
"I believe that all human life is precious, and as governor, I will work to ensure that the lives of the born and unborn are protected in Mississippi," Bryant said in his statement. "This bill requires all physicians associated with an abortion clinic in Mississippi to be board-certified or eligible in obstetrics and gynecology."