Hinrich hyperextended the elbow in the fourth quarter of Friday's loss to Phoenix.
Coach Flip Saunders said the team will wait a day or two for the swelling to go down before Hinrich has the MRI, making it unlikely the veteran will play in Monday's game at New York.
"We're more concerned about ligaments in the elbow," Saunders said, "because of how it was hyperextended."
Hinrich is averaging 11.1 points and 4.6 assists. His injury left rookie John Wall as the team's only point guard, so the Wizards on Saturday signed Mustafa Shakur to a 10-day contract.
Shakur was playing with the Rio Grande Valley Vipers of the D-League. He had previous 10-day stints with New Orleans and Oklahoma City but did not play in a regular season game with either game.
Saunders said Shakur would make his NBA debut against the Celtics, especially with the Wizards playing the second game of a back-to-back.
"We're hoping to give John some breathing room," Saunders said.