Seager was running between first and second base after hitting an opposite-field double into the left-field corner. He appeared to make the turn around first base cleanly, but then pulled up after a few steps and gingerly went to second base.
It was the fifth game in a row for Seager to have an extra-base hit. But after touching second base, he bent down and then almost immediately started going back toward the Rangers dugout, even before a team trainer got all the way out to check on him.
Manager Bruce Bochy said that Seager will get his hamstring reevaluated.
"There's some tightness there and we'll see where we're at [Wednesday]," he told reporters.
Josh Smith took over as the pinch-runner and remained in the game to play shortstop.
Seager is the second year of a $325 million, 10-year contract with the Rangers. He was 2 for 2 with a walk against the Royals and is hitting .359 this season.
Before playing 151 games in his debut with the Rangers last season, Seager missed about 2 1/2 months of the 2021 season with the Los Angeles Dodgers because of a fractured right hand after getting hit by a pitch. He spent nearly a month on the IL in 2019 with a left hamstring strain, and was limited to only 26 games in 2018 because of right elbow surgery.
The Associated Press contributed to this report.