Perisic has joined up with the rest of the Inter squad at their training camp in the north of Italy but United are trying to do a deal before the Serie A club flies to China next week.
Their most recent offer for Perisic is understood to have been £39m but Sky Sports News HQ understands they will need to increase that to around £45m to secure their target, who has three years left on his current contract.
He was due to meet with Inter boss Luciano Spalletti last week to talk over his future, with Spalletti saying: "You keep talking about Perisic and his desire to leave but I need to listen to his own words first and then decide how to act.
"I hope to have him in the side, I am counting on it and I am positive in that sense."
Perisic has been playing in Italy since the summer of 2015, scoring 18 goals in 70 Serie A games since joining them from German club Wolfsburg two games into the 2015-16 season.
The 28-year-old has also scored 16 goals in 57 games for his country.