The Rangers would give up reliever Chris Ray and a player to be named later and would also get some cash back so that neither team takes on additional salary.
FoxSports.com and USA Today first reported the trade.
The move will give the Rangers a right-handed hitting veteran catcher with a career .275 batting average. Molina, who turns 36 in July, is batting .257 with three homers and 17 RBIs in 61 games (202 at-bats) for the Giants this season. He is making $4.5 million this season.
The Giants can afford to do the deal because they can slip rookie Buster Posey behind the plate as the starter.
Rangers catchers were hitting .212 coming into Wednesday's game against the Los Angeles Angels, the lowest in the majors.
The position has been in flux since spring training, when Jarrod Saltalamacchia wasn't healthy to start the season. Matt Treanor is hitting .237 with five homers and 23 RBIs as the No. 1 catcher. Max Ramirez, his backup, is hitting .227 with two homers and eight RBIs.
Ray was 2-0 with a 3.41 ERA in 35 appearances (31 2/3 innings pitched), with 16 walks and 16 strikeouts.