Van Persie should be fit to return for the visit of Norwich City on April 26 or the trip to Sunderland a week later.
The Dutchman suffered a sprained knee after scoring a hat-trick against Olympiakos on March 19 to help United into the last eight of the Champions League, but has been out of action since.
Moyes is confident van Persie will prove his fitness ahead of leading his country in the World Cup in Brazil in the summer and said: "Robin is over in Holland having treatment.
"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."
Forward partner Wayne Rooney has been ruled out of Saturday's clash at Newcastle United and is a doubt for the return leg of the quarter-final against Bayern Munich.
Moyes is likely to turn to Danny Welbeck in Rooney's expected absence to lead the forward line in the second leg at the Allianz Arena on Wednesday, and the Scot added: "Danny has come in and done really well.
"He played well and we want to give some minutes to Chicharito as well because his opportunities have been limited due to Rooney and van Persie at times. We'll have to look at that as well.
"I think Danny has the fear factor. He has the ability to run away from people and has really strong upper-body strength, sometimes people don't give him the credit for that.
"His finishing has always been questioned but I think, this year, he's looked more likely. The year before, he only got two goals. This year, he's had a contribution and his goals have been important for us at different times.
"I'd like to think he has got better this year. Maybe people who have seen him more than me would have a better idea but I think he's got better and he has improved.
"He's one of these boys who can play as a wide striker. Football is becoming a little more flexible with how people play and not as rigid so you can't always title people and position them but Danny could be classed as a centre-forward or a wide striker as well."
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