Mourinho and United are on the hunt for a central midfielder and Nemanja Matic, Eric Dier, Julian Weigl and Naingollan are just some of the names in the frame.
But it seems the Red Devils will find it hard to lure Nainggolan away from Italy.
Asked at the press conference whether Nainggolan was staying put, he replied: "Yes (he will stay)". Now we build a team with options that the market provides. "I prefer to wait and bring in the player I really want, rather than to act hastily".
According to Italian sports daily Gazzetta dello Sport, the Nerazzurri are prepared to offer a fee of around €55m to €60m in order to tempt Roma into selling Nainggolan.
GETTY Manchester United target Radja Nainggolan in action for Roma
The club are believed to be stalling over a new deal for Nainggolan and during an Instagram live broadcast over the weekend, the Belgium worldwide was asked if he would sign a new contract to stay in Rome, to which Nainggolan replied: "I'm waiting, as always!"
Despite Monchi's comments, the Red Devils are looking to test the waters with a mega-money bid.
Sporting director Monchi insisted this week that they would not allow any more key players to leave the Stadio Olimpico but the former Sevilla man has come in for heavy criticism from supporters for allowing the club to cash in on their top players.
Roma are trying to tie their ace down to a new contract and bump up his current terms.