The bill's sponsor, Sen. Glenn Hegar, R-Katy, said he wanted to keep the 20-week restriction, but worried there wasn't enough time to pass it with just one week left in the special session.
Republicans are likely counting on the House to restore the 20-weeks limit.
No abortion bills made it to the Senate floor during the regular session, and Republicans had urged Gov. Rick Perry to add the issue to the special session agenda.
The bill would allow abortions only in surgical facilities and place greater restrictions on abortion-inducing medications.
Abortion rights groups complain the new restrictions could shut down 90 percent of the abortion clinics in the state and will reduce the effectiveness of medications.
The bill still needs a final Senate vote to send the bill to the House.
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