The Spain goalkeeper has grown disenchanted with life under the Dutchman, despite signing a new four-year contract at the start of the season.
UK media claim sources at Old Trafford say De Gea, who is wanted by Real Madrid, will press to make the move happen.
It promises to be a major issue for United.
De Gea has been their star player during a season that has seen them miss out on qualification for the Champions League.
And reports in Spain have claimed that Real are preparing to activate the 25-year-old shot-stopper’s £40m release clause written into his new deal last year.
Meanwhile, United are discussing whether to press ahead with a deal to sign free agent Zlatan Ibrahimovic, which the PSG star’s agent has denied.