Siemian, who came into the line-up after regular starter Sam Darnold was laid low with glandular fever, suffered ligament damage to his left ankle during Monday's loss to Cleveland.
It means the winless Jets will start Sunday's daunting divisional game against the New England Patriots with third-string quarterback Luke Falk.
Jets coach Adam Gase said however that the team was optimistic that first choice Darnold may soon return to fitness.
Darnold had reported to the Jets training facility on Tuesday and was on the mend, Gase said.
"He was in the building today," Gase said. "He's feeling better. The doctors are letting us know when to proceed with him."