French police had arrested a man suspected of the stabbing on Friday and he has now been detained in custody.
According to the Sky Sports television channel, the 44-year-old United fan was travelling in a taxi with friends to the centre of Paris after the game which the English side won 3-1 when the driver objected to their boisterous celebrations.
“The driver pulled over to get them out of his taxi, before pulling out a knife and threatening the female passenger in the group,” claimed Sky. “The victim tried to intervene to protect her, but the driver then turned the knife on him, stabbing him in the chest with what's been described as 'a large blade'.”
Manchester United sent their support to the fan.
“We were shocked to hear about the incident with one of our fans who was stabbed while in Paris for the Champions League game,” United said in a statement. “Everyone at the club wishes him well during his recovery.”
A French police source confirmed a stabbing incident to AFP after the game but said the victim was not believed to have suffered life-threatening injuries.