The bill`s author, state legislator Justin Humphrey, who has described pregnant women as "hosts", said it would give the father a "say-so".
Critics say it degrades women.
The fact Oklahoma lawmakers decided to move the measure toward a full vote by the legislature comes at a time the anti-abortion movement is growing bolder under President Donald Trump.
Mr Trump at one point suggested women should be punished for having abortions, and promised to elect an anti-abortion judge to the Supreme Court. The position of his nominee, Neil Gorsuch, on abortion is unclear.
The Associated Press says at least 11 anti-abortion bills have been introduced in Oklahoma this session, including one that classifies the procedure as first-degree murder, but it is unlikely many of those will reach the governor`s desk.
This particular bill would allow men to demand a paternity test before the procedure could be carried out, but does make exceptions for women who have been raped or are the victims of incest.
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