Jose Mourinho is a known long-term admirer of the 27-year-old, who has been sidelined with a serious knee injury since January 31.
And the mentioned source is claiming that the Portuguese manager is ready to offer Rose a lucrative deal, having missed out on his signature this summer.
Chelsea are also reportedly interested in securing the England international's services, but United remain front-runners.
Rose could make a long-awaited return to action later this month, but may find it hard to regain his place in the Spurs starting XI, given that Ben Davies has performed so well in his absence.