Alderweireld's future in North London remains uncertain, with a clause in his contract coming into effect next year which would allow him to leave the club for just £25m.
United were hopeful that they would be able to sign the Belgium international for £55m, but the Evening Standard reports that Spurs are holding out for a bigger offer.
Tottenham are reluctant to lose Alderweireld and have offered the defender a contract extension worth around £120,000 per week, but United's approach is understood to be significantly more lucrative.
The Red Devils are reportedly ready to offer the 29-year-old a five-year contract at Old Trafford as they look to bolster their defensive options this summer.
United's chances of prising Alderweireld away from Spurs could be aided by the fact that Tottenham are interested in signing Anthony Martial, who is thought to be considering his future at Old Trafford due to a lack of first-team football.