The Mirror reports that although United are preparing another offer for the Croatia international, their Premier League rivals are ready to hijack the deal.
The Red Devils have already failed to land the player with two bids, but latest reports claim executive vice-chairman Ed Woodward, is now close to meeting the £48million asking price.
Chelsea boss Antonio Conte is believed to be frustrated with a lack of new faces this summer, after bringing in only Antonio Rudiger, Alvaro Morata, Willy Caballero and Tiemoue Bakayoko.
Perisic enjoyed an impressive campaign in Serie A last season, contributing 10 goals and six assists. He has three years remaining on his current contract at Inter.