The Spain under-21 international has been heavily linked with both Arsenal and Spurs since the summer, and the report claims Tottenham have asked Real Madrid about his availability heading into the January window.
Any deal would only be on loan as the Spanish giants rate the 21-year-old forward highly and are keen to keep him at the club.
But he's struggling for first-team action this season - he's made just one start in La Liga - and Real Madrid feel he would benefit from minutes in the Premier League.
Tottenham tried to include Morata in the Gareth Bale transfer over the summer, but Real Madrid boss Carlo Ancelotti was reluctant to let the forward leave permanently.
Arsenal manager Arsene Wenger has subsequently been linked, with the Gunners looking to add to their attacking options next month.
Nicklas Bendtner is the only back-up to Olivier Giroud as an out-and-out striker, although the return of Lukas Podolski is a timely one.
Both north London clubs are expected to be busy next month, with both Arsenal and Spurs keen to bring in new faces for differing reasons.
Spurs continue to eye a top four spot and qualification for the Champions League, while Arsenal's impressive form means Wenger has his eye on a first title since 2004.
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