De Gea, who is out of contract next summer, has been heavily linked with a £50million move to Real Madrid.
The Spaniard is on Rafa Benitez's list of priorities, and he is keen to strike a deal.
Football news outlet Goal claim De Gea has his heart set on a move to Real.
For the moment there is a goalkeeper at United and we are watchful of his situation
Hugo Lloris
This is despite suggestions that De Gea's advisors have recommended the shot-stopper continue his development in the Premier League before considering a switch to La Liga.
However, it is unclear as to whether De Gea has handed in a formal transfer request.
United have been drawing up a shortlist of potential replacement, and Lloris appears to fancy a move to Old Trafford, as he said: "For the moment there is a goalkeeper at United and we are watchful of his situation.
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David De Gea could be off in the summer
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"You have to keep a certain calm with all this. If things should happen then they'll happen naturally."
Lloris has made no secret of his desire to play in the Champions League, even though he is contracted with Spurs until 2019.