The 32-year-old scored 10 Premier League goals for the Hornets during the 2019-20 campaign but was unable to prevent the club from dropping into the Championship.
According to the Daily Mail, Spurs are keen on a loan move for the experienced forward, but Watford are ready to offload him permanently following their relegation to the second tier.
The report claims that Mourinho, who is seeking cover for Harry Kane, is an admirer of the Englishman.
Deeney only has one year left to run on his contract at Vicarage Road and could therefore be available for a relatively small fee during this summer's transfer window.
The Birmingham-born attacker has scored 133 goals in 398 appearances for Watford in all competitions.