The Wales manager has urged supporters to get behind the Portuguese boss and insists he does not see a better alternative than the 55-year-old.
Manchester United have lost two out their first three league games including a heavy loss at home to Tottenham on Monday.
The run of losses has seen the pressure mount on Mourinho, with the likes of Zinedine Zidane, Antonio Conte and even Giggs himself having been linked with replacing the 55-year-old .
When asked if he is interested in replacing under-fire Jose Mourinho, Giggs said: “Manchester United have got a fantastic manager already there.
“The club are going through a tough time but you’ve got to come through those tough times.
“I don’t see where Manchester United go after Jose Mourinho. I think they’ve got the right man and they should stick with him.
“I think purely coming from me being a United fan, I was there on Monday [for the Tottenham defeat], I think it’s a time for the fans to stick together, get behind the club, get behind the team, get behind the manager, get behind the coaching staff.
“That’s me purely speaking as a fan.
“I think it’s a tough time at the moment but we’ve had tough times before and we’ve come through it.”
Manchester United will hope to return to winning ways on Saturday with a visit to Burnley at the Turf Moor Stadium