There had been hope for a return Sunday against the Philadelphia Eagles, but Darnold -- diagnosed three weeks ago with mononucleosis -- wasn't fully cleared by doctors after another round of tests on Monday.
Darnold received partial clearance, meaning he can resume throwing and do non-contact work, but his availability for Sunday is "a question mark for us," coach Adam Gase said.
It means the Jets (0-3), who have scored only one touchdown from scrimmage, may have to start former practice-squad quarterback Luke Falk for the second straight game.
Gase acknowledged Darnold's test results were "not the clear-cut answer we were hoping for ... There's just a lot of gray area right now."
Darnold deferred to Gase when asked if his spleen remains enlarged, and Gase wouldn't give a direct answer.
This is a setback for the Jets. Coming off the bye, they had been hoping to rally around Darnold's return in an attempt to re-boot their bitterly disappointing season.
Gase didn't rule out Darnold for Sunday, but he didn't sound optimistic.
"We have to see how practice goes, we have to see how he feels, we have to see what his conditioning level is," Gase said. "There are a lot of boxes we have to check before we can say, 'Hey, we're rolling on Sunday.'"
Because of Darnold's uncertainty, the quarterback situation for the week is "not ideal," said Gase, noting that the inexperienced Falk needs a full slate of practice reps.
The most likely scenario for Sunday is Falk starting, with former Dallas Cowboys draft pick Mike White -- signed last week to the practice squad -- being elevated to the No. 2 role.
Darnold said he's "super anxious" to play, but he hasn't done any heavy cardio work yet. He's been limited to walking and using a stationary bike, meaning his conditioning could be an issue.
"I feel 100 percent," Darnold said. "I've been feeling 100 percent for a week now. That's the most frustrating part about this."
Darnold said he feels well enough to play this week, but he acknowledged he hasn't practiced in three weeks and that timing with his receivers could be a problem. It's also worth noting he still hasn't been cleared to lift weights, which illustrates he still has a ways to go.
He was under the weather in the Jets' season-opening loss to the Buffalo Bills and was diagnosed two days later with mono. The biggest concern with the illness is an enlarged spleen, which could rupture with a blow to the midsection.
"My spleen at this point is going to do what it's going to do," said Darnold, who, for the first time since the diagnosis, evaded questions about the status of his spleen.
The Jets can't catch a break at quarterback. Darnold was replaced by Trevor Siemian, who suffered a season-ending ankle injury. In six quarters, Falk has completed 68 percent of his passes (32-for-47) for 296 yards and one interception, but he has led the offense to only three points.
In other injury news, middle linebacker C.J. Mosley could miss his third game with a groin injury. There had been hope that he could return after the bye, but he's not expected to practice Wednesday. Mosley said he can run straight ahead, but he hasn't started cutting yet.
Defensive tackle Quinnen Williams, drafted third overall, is expected to return after missing two games with a sprained ankle.