Morata is expected to leave Real Madrid with Chelsea and Manchester United both being heavily linked with him.
Express Sport understands the Spain international is on the Blues' wishlist to replace Diego Costa, who could leave for the Chinese Super League.
United, meanwhile, have reportedly upped their interest in the striker with top target Antoine Griezmann set to stay at Atletico Madrid and Zlatan Ibrahimovic expected to leave for a new challenge elsewhere.
AC Milan are also in the mix for his signature but their failure to secure Champions League football could count against them.
Alvaro Morata's next club: Where could the Real Madrid ace go next? Tue, May 16, 2017
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Where could Alvaro Morata go next? Odds provided by Paddy Power
Real Madrid star Alvaro Morata would prefer to join Chelsea over Manchester United
And Spanish outlet Diario Gol are saying that, out of all three teams, it is Chelsea who Morata wants to join the most.
The 24-year-old played under current Blues boss Antonio Conte during their time together at Juventus, winning the Serie A title in the process.
It is said that Morata wants a reunion at Stamford Bridge – despite the fact United manager Jose Mourinho gave him his Real debut.
Manager Zinedine Zidane looks likely to cut him loose after tonight's Champions League final, despite an impressive goal return of 20 goals from 42 games.
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Alvaro Morata looks set to leave Real Madrid with Manchester United and Chelsea keen
Zidane has often picked the inconsistent Karim Benzema ahead of Morata and the player looks set to leave in pursuit of regular first-team football.
However, the report adds that Chelsea will have to raise their offer if they are to lure the striker away from the Bernabeu.
While Milan are reportedly willing to pay £60million for him, the Blues are supposedly only willing to pay £35m.
United have the finances to broker a deal but the news that Morata would prefer a move to Chelsea instead represents a major blow.