And the Manchester Evening News claim United are prepared to wait until deadline day for a deal to be completed.
However, it is thought this depends on Saints softening their stance towards the player.
Jose Fonte - 5 things you need to know
Reports elsewhere suggested Southampton have offered Fonte a new two-year deal, so as to keep United at bay.
United have so far spent £145m on Paul Pogba, Zlatan Ibrahimovic, Henrikh Mkhitaryan and Eric Bailly.
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