The versatile defender or midfielder suffered an ankle injury during Monday's FA Cup fifth-round win over Reading and is struggling to be fit for the Champions League last-16 second-leg match against Real Madrid on March 5.
"Phil Jones is the one we are really concerned about in terms of the Real Madrid game," manager Alex Ferguson told reporters.
"We're working hard on it and I'd say he has an outside chance - no more than that. He won't be fit for the weekend."
Jones man-marked Real forward Cristiano Ronaldo during the 1-1 draw in the first leg in Madrid.
"He did an incredible job against Real Madrid," Ferguson said.
"The great thing Phil has got is that he doesn't care who he is playing against. There is no fear about playing against anyone. We will have to wait and see. The Madrid game is two weeks away. A lot can happen in that time."
United travel to bottom club QPR 12 points clear at the top of the Premier League with 12 games left.
"QPR are fighting for survival and we know it is going to be a hard game," Ferguson said. "That is what we need anyway. We need to keep the awareness on making sure we do our job right."
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