The 32-year-old confirmed earlier this month that he will leave Old Trafford at the end of the campaign, with Inter among the front-runners for his signature.
While United have endured a disappointing campaign in David Moyes' first season at the helm - sitting sixth in the Premier League, 15 points adrift of leaders Chelsea - they remain in contention for European glory.
Ahead of the Champions League last 16 first leg with Olympiacos, Vidic asserted that United could do "something special" in the competition.
"I believe that if we keep playing as we did (in Saturday's win over Crystal Palace) - solid - and improve our form again (then) we have a chance to do something good for the rest of the season," said Vidic in quotes reported by the British media.
"That means fourth place and the Champions League.
"I believe we can do something special in the Champions League. I really do."
United ended a run of three games without victory on Saturday - Robin van Persie and Wayne Rooney helping the champions to a 2-0 win at Selhurst Park.
And Vidic believes there have been signs of improvement ahead of their trip to Greece, as he aims to win the Champions League for a second time with the club.
"We always believe we can do well, but we know we have to improve in every game and I think we saw some signs (at Palace)," he added.
"To win it you have to defend well as a team. This is the way you should play and that is encouraging for us before we go to Greece."
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
‘He’ll still be playing at 40 at a good level because he’s in such good shape and looks after himself so well. He does everything to be a top professional’: Ex-Liverpool coach insists Mohamed Salah has plenty more miles in the tank
