The measure, which passed the House Public Health Committee on Tuesday, would allow abortion only to save a woman's life. It would make no exception in cases of rape or incest.
Barbour, a Republican, said he preferred an exception in cases of rape and incest, but if such a bill came to his desk: "I suspect I'll sign it."
The full House could vote on the bill next week, and it would then go to the Senate.
South Dakota lawmakers passed a similar bill last week that was intended to provoke a legal showdown over Roe v. Wade, the 1973 Supreme Court ruling establishing the right to an abortion. The measure is awaiting Republican Gov. Mike Rounds' signature. He has said he is inclined to sign it.
Mississippi already has some of the strictest abortion laws in the nation. It requires a 24-hour waiting period and counseling for all abortions, plus the consent of both parents for minors who seek the procedure.