The Brazilian star was reduced to tears as he limped off the pitch during last Wednesday's French Cup win over Strasbourg.
According to L'Equipe, the forward will be stuck on the sidelines until at least the end of March.
He is set to miss the first leg at Old Trafford on February 12 and the return fixture in France on March 6.
Neymar will also be absent for league games against Lyon, Bordeaux, St-Etienne, Montpellier, Nimes and Caen.
The striker has been in sublime form this season, with 18 goals and eight assists for the Ligue 1 leaders.
United, who have won eight games on the bounce since Ole Gunnar Solskjaer replaced Jose Mourinho last month, have been given a huge boost ahead of the last-16 tie.
PSG boss Thomas Tuchel admitted after Tuesday's match that it wasn't looking good for the 26-year-old.
He said: "It's too early for Neymar. He has started getting treatment, we need to wait until next week.
"His treatment will depend on how the injury develops in the next few days. The fact Neymar and Marco are injured is a sad situation for us.
"We need to find solutions, collectively, so that we carry on winning."
Neymar fractured his metatarsal after a heavy challenge from Anthony Goncalves and was taken to hospital after the match.
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