Grilli was 2-4 with a 7.97 ERA and one save in 26 games with Toronto this year before being designated for assignment. Grilli, 40, was an All-Star in 2013 when he had 33 saves for Pittsburgh. He also recorded 24 saves for Atlanta in 2015.
He was expected to join the Rangers on Monday when they return to Texas from Chicago, where they were facing the White Sox on Sunday.
The Rangers said they would make a corresponding move on Monday to clear room on the 25-man roster. The Rangers designated right-hander Preston Claiborne, who is on option at Triple-A Round Rock, to make room on the 40-man roster for Grilli.
Pinto is hitting .311 with four home runs and 18 RBIs in 46 games for Class A Down East in the Blue Jays’ minor league system. The left-handed hitter is in his sixth professional season after being signed as a non-drafted free agent.