Rangers general manager Jon Daniels was hopeful the club's sluggers wouldn't be out for long, but timetables for their return were unclear.
Hamilton left Tuesday's 11-3 loss to the Angels in the fourth inning with what the Rangers called "sinus issues."
Hamilton had a sinus MRI Wednesday evening and the Rangers are expected to know the results Thursday.
The major league leader in home runs, Hamilton was also having his vision checked by a specialist Wednesday.
"(A sinus problem) was the initial, quick read based on the symptoms he was feeling," Daniels said. "He's had issues with his vision and focus, and we thought it was related to a sinus infection. Seeing a doctor earlier today, there hasn't been any indication that there is a sinus infection to this point.
"I don't know that we have an appropriate term to use at this point, but the symptoms are consistent with what somebody might feel having a sinus infection."
Neither Hamilton nor Beltre were at the ballpark Wednesday. Beltre had intestinal issues Tuesday but played through them.
"He was in some discomfort, pushed through it but continued to have those feelings today," Daniels said. "We're going to get some additional tests and make sure it's not anything significant.
"If Adrian could play, he'd be out there. When we say intestinal it's more than a stomachache. Hopefully we'll get to the bottom of that and he'll be back out there (Thursday)."
Hamilton, the Rangers' No. 3 hitter, entered Wednesday leading the majors with 42 home runs and is second with 123 RBIs. Beltre, the cleanup hitter, entered Wednesday fifth in the AL with a .316 batting average to go with 33 homers and 93 RBIs.
The Rangers have a four-game lead over the Oakland Athletics and a 7½-game lead over the Angels in the AL West.