The deal is pending a physical, and neither team confirmed the trade. It is not yet clear which prospects the Giants will get in return for Moore.
Moore, 28, was acquired by the Giants in 2016 at the trading deadline in a deal that sent infielder Matt Duffy and two minor leaguers to the Tampa Bay Rays.
Moore went 6-5 with a 4.08 ERA in 12 games for the Giants in 2016, but he was just 6-15 with a 5.52 ERA in his first full season with the Giants in 2017.
The Giants exercised their $7 million option on Moore last month, and he is scheduled to earn $9 million in 2018 and has a $10 million club option (and $1 million buyout) for 2019.
The Giants were unable to trade for Giancarlo Stanton, and they now appear to be cutting payroll as they seek a third baseman and outfielder.
The Rangers are expected to add Moore to a starting rotation that includes Cole Hamels, Mike Minor and Martin Perez, all of whom are left-handed. Right-hander Andrew Cashner, an 11-game winner, became a free agent after the season.