There had been speculation throughout baseball that Lee would be traded to the New York Yankees. Lee had been scheduled to pitch against the Yankees Friday, but was scratched from the lineup during the afternoon.
Lee, 31, was in his first season with Seattle after pitching for both Cleveland and Philadelphia in 2009. He is 8-3 this season with a 2.34 earned run average in 13 starts and has won his last four outings. Lee has struck out 89 batters this year and walked only six.
"He has the ability to pitch to both sides of the plate, has an outstanding breaking ball and likes the big games," said Nolan Ryan, president of the Rangers.
Texas began Friday's action with a 5 1/2-game lead over the Los Angeles Angels in the American League West. The Rangers are seeking what would be their fourth playoff appearance and are still trying to win a post-season series for the first time.
Smoak was a first-round pick in 2008 and was called up this season to be the team's regular first baseman. He is batting just .209 with eight home runs and 34 RBI in 70 games.