The 30-year-old Netherlands international striker is sidelined for up to six weeks after spraining his knee during the Red Devils' 3-0 victory over Olympiacos in the second leg of their Champions League last-16 tie at Old Trafford last month.
"Robin is over in Holland having treatment," United's official website quotes the Scot as saying. "We get a daily update on how Robin is and he's getting better. Time is healing it as well and he's doing a lot of rehabilitation work and recovery back there.
"He's coming on, as far as we know. I still see it as four to six weeks from when he got the injury. That was what we were told from the doctors and there's not been any reason why it would be any more time or less time. But we'll monitor that. My hope was we'd get him back before the end of the season for sure. I think the big thing is we will be medically led.
"If he's not right, we certainly wouldn't bring him back too soon. We'll be totally led by the medical team and Robin. He's a good judge of how his body feels and how we've tried to deal with Robin all through the season is we have given him the understanding that he tells us how he feels. He'll be the first one to tell us how he is."
Van Persie has scored 11 goals in the Premier League and four in the Champions League so far this season.