Morrison, 21, was a key performer for the Hammers earlier in the season, even as the team struggled to get their campaign up and running.
His form led to England Under-21 recognition but injuries, a slump in performances and the return to fitness of a number of West Ham's key players have seen Morrison's path back into the first-team fold restricted.
A move to fellow Barclays Premier League strugglers Fulham in January failed to materialise and Allardyce had to refute claims Morrison had threatened to strike to force through a deal.
Now a loan move to a Sky Bet Championship side appears to be a distinct possibility, with QPR keen and with Allardyce frustrated at how long Morrison's injury has kept him on the sidelines.
"Listening to our medical staff, there is no problem that he can carry on - not at all," he said.
"He says he feels it occasionally. Instead of gritting your teeth and getting on with it... he's not the type.
"Lots and lots of players throughout the country will be playing with a similar type of injury that the medical team say 'you can carry on, it's not a problem'."
Back-to-back wins have seen the Hammers move up to 15th in the Barclays Premier League after a dismal couple of months, with Allardyce left bemoaning a fall in impact from former Manchester United youngster Morrison.
"Some come, rise to the occasion and then don't realise how far they have risen, then all of a sudden when they start realising where they have gone that can be a little bit of a problem," he added.
"I think that overall anybody under 21 playing in the Premier League on a regular basis finds it very difficult eventually. I think probably (Manchester United winger) Adnan Januzaj is the one who has played more. Luke Shaw would probably be the other one at Southampton.
"You have a lot of other players that are challenging his (Morrison's) position and doing a better job than him. I'm always disappointed if no one gives me selection problems.
"I like the fact that you have to see how players deal with being left out - are they going to be professional or unprofessional about it?
"I'm not talking about Rav as an individual, but everybody, because some players won't even get on the bench shortly. It will be interesting for me to see their professionalism and determination to react the right way, or (whether) they go away and sulk."
Allardyce refused to answer any further questions on Morrison at his press conference on Monday morning, with rumours surfacing shortly afterwards that Morrison was set to move away on a temporary basis.
QPR boss Harry Redknapp told Sky Sports News after his side's loss to Derby: "It's a possibility, I've been trying to speak to Sam. I think he'd like to come and if they want to loan him, we're here."
Morrison will not form part of the West Ham squad that will entertain Norwich on Tuesday evening looking to strengthen their chances of survival with a third successive league victory.
Andy Carroll will also be missing as he serves the second of a three-game ban but Allardyce will be able to call on Joe Cole and Mohamed Diame after the midfield duo returned from injury.