Fans have been asked to submit their entries on McCartney's YouTube page as a response to the track's official clip by June 16.
The entrant with the most 'likes' by viewers on YouTube is to be crowned as the competition's winner and will receive a deluxe editions of the newly re-mastered McCartney and McCartney II albums as well as Wings' Band on the Run as a prize.
McCartney recently revealed that he is in the midst of recording an LP of pop standards as a tribute to his late father James.
He will also join John Lennon's widow Yoko Ono and the late George Harrison's wife Olivia at June's fifth anniversary for the Beatles-inspired Love stage show in Las Vegas.
In May, McCartney became engaged to long-time girlfriend Nancy Shevell. The singer has been married twice before.
> Rolling Stones 'envious of Beatles' singing'
Watch Paul McCartney's 'Maybe I'm Amazed' music video below: