West Virginia Gov. Earl Ray Tomblin (D) vetoed a similar measure last year, although Republican pickups in the 2014 elections give antiabortion advocates the opportunity to override a veto.
The South Carolina measure passed the state House last year, but it died when the Senate didn’t take it up in time. Gov. Nikki Haley (R) has said she would sign such a bill.
The measures are similar to the federal ban, called the Pain Capable Unborn Child Protection Act, that Republicans had to pull from the floor of the U.S. House last month.
Legislators in Wisconsin are working to fashion their own version of the 20-week ban.
Abortion rights supporters say the pushback that forced congressional Republicans to drop the 20-week abortion bill is a preview of what’s to come if states take up the same measure.
Members of Congress, including Reps. Renee Ellmers (R-N.C.) and Charlie Dent (R-Pa.), warned that taking up legislation just a few weeks into the new Republican majority would revive uncomfortable associations with arch conservatives like former congressman Todd Akin (R-Mo.) and others.