Spurs were initially reluctant to do business with one of their leading Premier League rivals, unless they got a deal for Anthony Martial in return.
Spurs were also demanding a high price for Alderweireld, but they have struggled to offload the Belgian.
With the transfer window now edging closer to shutting and Alderweireld still at Spurs, Levy has realised that keeping the centre-back around may not be the best option.
And Spurs are now said to be far more receptive to United’s offer for the 29-year-old.
United, though, have other targets - with Harry Maguire now strongly linked.
It remains unclear whether Jose Mourinho’s side will come back in for Alderweireld, but if they do, it seems that Spurs would be much keener to do a deal with the Red Devils at this stage of the summer.