Coach Bruce Arians said Wednesday that although Palmer has made "great progress," he will not travel with the team to the Bay Area.
"There's no way I'm going to take any chances with him," Arians said. "He argued a little bit, but he is not cleared. But he's getting much better and feels fine. Hopefully we'll have him back next week."
The Cardinals, who host the New York Jets on Monday Night Football on Oct. 17, will have 10 days between games, leaving Arians "very optimistic" that Palmer will be healthy by then.
Arians said Palmer was not experiencing headaches anymore.
"He actually seemed fine this morning," Arians said. "It's just a matter of going through the tests."
Palmer's absence leaves Drew Stanton as the Cardinals' starter. He went 5-3 as a starter in Palmer's place in 2014, but he hasn't started a game since. He's 5-for-15 this season for 48 yards and two interceptions, both coming Sunday after he replaced Palmer.
Stanton struggled with high passes during training camp, but Arians said Stanton has resolved an undisclosed knee issue and has "worked hard" at getting his passes under control.
The Cardinals also promoted quarterback Zac Dysert to the active roster from the practice squad on Tuesday. Arians said Dysert has a separate package specifically designed for him.
"He's a very bright guy," Arians said. "He has a package he knows very well and is very comfortable with. We'll use that package if we have to use him."