Gillespie reckons Sancho wants to leave Borussia Dortmund and complete a move to Old Trafford to become a "world superstar".
"He is a fantastic player and still very young," Gillespie told Sky Sports.
"He has done fantastically well at Dortmund and obviously wants to come to the Premier League.
"No disrespect to Dortmund they are a massive club as well, but they are not Manchester United.
"Joining a club like Man Utd can help Sancho go on to become a world superstar.
"With what he has been able to do at both Dortmund and England, fingers crossed it is a transfer that might go through."