It is claimed Zinedine Zidane’s side are primed to make a renewed approach to sign De Gea for a world-record fee for a keeper of £65m.
Oblak is being heavily considered as a direct replacement should United lose their star shot-stopper.
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Manchester United's David De Gea is set to cost Real Madrid a world-record fee for a keeper
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Mourinho has apparently ordered regular scouting checks on the £40m-rated 24-year-old.
He already has the support of Griezmanm, who is said to rank Oblak as the best keeper in the world, ahead of the likes of De Gea, Thibaut Courtois and Gianluigi Buffon.
Manchester United are already planning to trigger the France international’s £85m release clause, and are keeping a close eye on Oblak too.