A state Senate committee is scheduled to vote Thursday on whether to send its version of the bill to the full Senate.
The legislation would ban abortions past 20 weeks of gestation, require abortion clinics to become ambulatory surgical centers, tighten usage guidelines for the drug RU486 and require doctors who perform abortions to have admitting privileges at a hospital within 30 miles of the clinic at which they're providing abortion services.
The House's move comes two weeks after state Sen. Wendy Davis tried to block the bill with a filibuster, talking for more than 10 hours in an attempt to run out the clock on the legislative session.
Gov. Rick Perry called a special session so the legislature could take up the measure again.