United manager Jose Mourinho had identified the England international as his primary defensive midfield target and was keen on securing him ahead of next season.
The Red Devils were ready to test Spurs' resolve by offering their desired valuation of £50m ($64.7m) for Dier, but Tottenham manager Mauricio Pochettino is keen to keep him at White Hart Lane and has asked the club's hierarchy to reject any approach from United.
Dier is said to be open to a move to Old Trafford but Tottenham are unlikely to allow him to leave and strengthen a direct rival.
The North London club's desire to hold on to their player seems to be working as, according to the Daily Mail, United are ready to make an approach for Nainggolan, who will cost less than the English midfielder.
The report claims that United can complete the deal for around £40m and it is believed that the midfielder would be open to leaving the Serie A outfit this summer.
The Belgian midfielder has been a key player for Roma in recent seasons, but new sporting director Monchi is said to be ready to offload him to use the funds to rebuild the squad.
Inter Milan have also shown interest in Nainggolan, but Roma are not interested in selling to their direct rivals. However, they are open to negotiating with United over a summer move.
Mourinho is an admirer of the midfielder, who was on Chelsea's radar last summer, and he is likely to be an able alternative to Dier, who was the Portuguese manager's preferred target.
The 29-year-old played in an attacking midfield role last season but can also play in a more central or defensive role. He contributed with 11 goals in his 37 Serie A appearances.
A midfielder is high on Mourinho's priority after the additions of Romelu Lukaku and Victor Lindelof. Apart from Nainggolan, the manager is also keeping tabs on Nemanja Matic, who could become available if Chelsea sign Tiemoue Bakayoko from AS Monaco.