According to the Mirror, the Serbian international has expressed his desire to reunite with Jose Mourinho at Old Trafford. The Red Devils current manager was in charge at Stamford Bridge, when he re-signed the midfielder for the Blues in January 2014.
Matic has played a key role in Chelsea's last two title winning campaign. He played instrumental role alongside N'Golo Kante as the west London club won the league in Antonio Conte's debut season.
Chelsea completed the signing of Tiemoue Bakayoko from AS Monaco earlier this summer. The arrival of the France international has seen the former Juventus and Italy manager sanction his sale.
The report claims United have made an opening bid of £35m ($45.5m) for Matic. However, the Premier League winners are demanding a fee of £50m ($65m) in letting their midfielder swap the Blues for the Red Devils.
Chelsea owner Roman Abramovich is ready to sanction Matic's sale to United as a goodwill gesture for Matic's five-and-a-half year service across two spells for the club. The Russian billionaire intervened to allow Petr Cech join Arsenal, despite Mourinho, their then manager's refusal for the transfer.
United managed to beat the English champions in securing the services of Romelu Lukaku from Everton. The Belgium international was all but set to return to his former club, before the 20-time English champions hijacked the deal from Chelsea.
Abramovich wants to add "Lukaku tax" and thus the asking price for United has been increased to £50m for Matic. The former Benfica man is also wanted by Juventus and Inter Milan and they could sign him for a lower fee.
However, Matic has set his sights on joining United this summer. The midfielder and Diego Costa are left of Chelsea's squad for their pre-season tour of China and Malaysia.
Meanwhile, Mourinho has urged United executive vice-chairman Ed Woodward to complete the third summer signing "as soon as possible." So far, the Red Devils have completed deals for centre-back Victor Lindelof and Lukaku. It should be seen whether Matic can be their third summer signing.