The 50-year-old Scot will officially take charge at Old Trafford on July 1 after Sir Alex Ferguson called time on his 26-year reign at the club at the end of the last campaign.
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"It was a surprise to us all that Ferguson left," De Gea told reporters.
"But we know that when we get back to training he [Moyes] will be there and it is going to be a pleasure to work with him."
Spain have already secured a place in the semi-final of the European Championship with one group match remaining following wins against Russia and Germany.
"We are into the semi-finals again and we are very happy with that," said De Gea.
"They are all good teams when you get to this stage of a tournament, they all have a lot of quality.
"It will be difficult whoever we face. We have had two tough games and, thankfully, we have been lucky enough to get through."