Anti-abortion bills advancing in several state legislatures appear to have a far better chance of passing than in past sessions -- a development that could further complicate the country's patchwork of local laws and create a scenario where more women search across state lines for certain abortion services.

Though most states have restrictions against late-term abortions, two states moved bills this week that would make those restrictions tighter. The Kansas Senate passed a bill Wednesday prohibiting most abortions after the 21st week of pregnancy. At the same time, the Idaho Senate backed a similar proposal with a 20-week threshold.

The bills were modeled after legislation passed last year in Nebraska premised on research suggesting a fetus is able to feel pain after 20 weeks. Similar proposals are percolating in at least nine other states, according to one organization's estimate...