But critics say the move would violate the constitution and affect children in need of stable homes.
"This goes against the best interest of the child,” said Catherine Oakley, senior legislative counsel for the Human Rights Campaign.
"This would allow adoption agencies to turn away qualified, loving parents - who are perhaps perfect in every way - because the agency has a difference in religious belief.
"As a governmental entity, Texas is bound to treat people equally under the law. This is a violation of equal protection under the law."
State Republican James Frank, the bill's author, said he wanted to make "reasonable accommodations so everyone can participate in the system."
"Everyone is welcome. But you don't have to think alike to participate," said Mr Frank, a Republican from rural Wichita Falls, near the Texan border with Oklahoma.
"My guess is if you have an LGBT agency they're going to pick an LGBT family, and if you have a Baptist agency they may be more likely to pick a Baptist family. They're free to do that and should be free to do that."
