Jose Mourinho has set his sights on signing the two attacking players and the Red Devils are preparing a £160m ($200m) move in landing Griezmann and Lukaku, reports The Sun. The Portuguese tactician currently has the services of Zlatan Ibrahimovic to lead United's attack.
The former Juventus and Barcelona's striker's contract at Old Trafford runs down after the end of the season. United have an option to extend it by a year and Mourinho is keen on keeping the tall Swede at the club for the next season.
Griezmann has long been identified as a transfer target for the 20-time English champions. Mourinho is in no mood to give up, despite strong interest from top clubs from England and across Europe for his signature.
The France international's advisor Eric Olhats has also confirmed the striker is also wanted by United's local rivals Manchester City, Mourinho's former club Chelsea and Atletico's league rivals Barcelona and Real Madrid.
"Chelsea, Manchester City, Barcelona and even Real have knocked on the door to see where things are, everyone comes to see what is do-able, and what is not do-able, that's normal," Olhats told RMC.
United signed Paul Pogba from Juventus for a world-record fee of £89.3m and Mourinho will tell his employers to match the fee if that is what is needed to beat competition from other clubs in signing Griezmann.
Meanwhile, Lukaku is also edging closer towards leaving Goodison Park after the end of the season. He has also informed Everton manager Ronald Koeman that he has decided against signing a new contract.
The Belgium international is attracting a strong interest from Chelsea and Mourinho remains hopeful he can beat the competition from the Blues in re-signing his former player. It was the current United boss who sold Lukaku to Everton in 2014, when he was in charge at Stamford Bridge.
Lukaku is earning £120,000-a-week ($150,264 per week) at Everton and Mourinho is willing to double his wages, if he decides to swap the Merseyside club for United. Any move for the Premier League top scorer will see the Old Trafford club splash around £70m ($87.6m).
United's hope of signing Griezmann and Lukaku rests on their chances of qualifying for the next season's Champions League. They can either achieve the feat by finishing the season in top four or by winning the Europa League.