Robertson declined his $7 million mutual option for 2025 on Saturday, Evan Grant of The Dallas Morning News reports.
Robertson picked up the save Thursday against the White Sox, striking out two batters over two perfect innings.
Robertson allowed one hit, hit a batter and struck out three over a scoreless eighth inning, earning a hold in Thursday's 3-1 win over the Dodgers.
Robertson earned his 13th hold in Wednesday's 3-2 win over the Dodgers. He struck out all three batters faced in the eighth inning.
Robertson (2-1) was charged with a blown save and was the losing pitcher in Saturday's 8-3 loss to the Rockies. He allowed four runs, two hits, a walk and hit a batter without getting an out.
Robertson gave up a run on two hits and a walk over 1.1 innings Wednesday, getting charged with his second blown save of the season in a 5-4 win over the Tigers. He struck out one.
Robertson worked a 1-2-3 eighth inning in Friday's 12-8 win over the Astros.
Robertson worked around two hits while striking out one over a scoreless inning Wednesday against the Rays.
Robertson is poised to begin the season in a setup role after Rangers manager Bruce Bochy said Sunday there is a "strong possibility" Jose Leclerc will open as the closer, Evan Grant of The Dallas Morning News reports.
Robertson (6-6) earned the win Sunday against the Phillies. He allowed a hit and two walks while striking out three in a scoreless inning.