The Belgium international has reportedly rejected a new offer of around ÂŁ120,000-a-week from the north London club.
And, according to The Evening Standard, United boss JosĂ© Mourinho is keen on making a move for the defender.
It was reported on Tuesday morning that, along with four other first-team players, Spurs would consider listening to offers for Alderweireld.
Paris Saint-Germain and Chelsea are also supposedly keen on the 29-year-old, but it is the Red Devils who are confident of securing a ÂŁ40 million move.
Daniel Levy is said to be reluctant to offer the player such a lucrative contract to take him into his mid-thirties â€“ something United would have no qualms about doing.
Fitness issues haveÂ limited the Belgian’s contribution this term
Alderweirled is now on holiday in Greece, so Mauricio Pochettinoâ€™s hopes of continuing contract talks are currently on hold.
One player who has signed a new contract with Spurs is Colombian defender Davinson Sanchez.
The former Ajax defender, who arrived for a club-record fee of ÂŁ42 million last season, has signed a new and improved six-year deal with the Londoners.