The coffee giant is hoping The Beatles legend will be the first artist to sign to Starbucks Records, their fledgling record label.
A source told Britain's The Sun newspaper: "All the major record labels have been bidding to sign Paul's next album - but it looks like they will all be caught out by the Starbucks deal."
"It's much more forward-thinking and creative than the deals the established labels put forward to Paul."
Starbucks plan to sign, record and produce its own artists and also start a digital music download site with another company.
The company already has a music venture called Hear Music, which releases compilations featuring tracks that have influenced established musicians.
The Rolling Stones and Sheryl Crow have previously selected their favorite songs for the Artists Choice compilations.
Hear Music was involved in releasing the late Ray Charles' final album "Genius Loves Company" in 2004 - which went on to win eight posthumous Grammy awards.
Paul, who has worked with Capitol Records on and off for 43 years, will take his entire back catalogue of Wings and solo work with him if he signs to Starbucks Records.
