Inter fly to China on Tuesday to begin their pre-season tour, with United keen to seal a deal before Perisic goes off with the rest of the squad.
United are said to have offered £39.5m for Perisic, plus £4.5m in add-ons, taking the total package to £44m for the Croatia international.
Inter are reluctant to lose Perisic, but United are hopeful the Italian club will find their offer for the winger impossible to turn down.
According to reports, the Nerazzurri are looking at Lazio star Keita Balde as a possible replacement for Perisic, should he move to Man United.
Balde has just one year to run on his contract with the Biancocelesti and had been thought to want Juventus above other interested clubs.
Chelsea and Liverpool also registered an interest in the €20m-rated forward, but Inter appear to be the front runners, which could allow the Serie A giants to release Perisic.
Inter Milan manager Luciano Spalletti recently appeared to drop a hint following reports that United had agreed a fee with Inter.
Speaking to Sportitalia, Spalletti said: “Perisic? You [journalists] know everything, it’s all out in the open.”