Price, 20, started the LV= Cup win over London Wasps for his senior debut but was injured just half an hour into the match.
The second-row spent almost 10 minutes receiving treatment on the pitch from both Tigers and Wasps medical staff before being stretched off in a brace protecting his neck.
Tigers director of rugby Cockerill said: “He is fine. He went to hospital at the Leicester Royal Infirmary across the road.
“They were exceptionally good with him and we had all of the right scans and X-rays.
“I say thankfully it is just concussion, but from the look of the injury at the time we are grateful that it is just concussion.
“He is recovering from that. He is doing pretty well and hopefully in the next week or two he will be fit to play again.”
Cockerill reported that Croft had taken a knock and there was no connection with the neck injury which sidelined him for nine months before his return to action three weeks ago.
“Tom has bruising in his back,” said Cockerill. “We are hoping he will take some part in the weekend (at Scarlets) and will play a little bit of time because that’s important, but we still have to manage his neck issue and the other things around that.”