The 36-year-old, who has become second fiddle to keeper Brad Guzan at Villa Park, is believed to have left the mansion he shared with wife Jane and moved to a smaller property outside of Wilmslow, Cheshire.
"Shay and Becky have been seeing each other for a while and have been meeting up more and more often recently," a source told The Mirror.
"She is a gorgeous nightclub PR girl and it is easy to understand what he sees in her. They met some time ago at a football event but it is only recently after Given split up with his wife Jane that they became a couple."
Gibson was previously in a relationship with League Two's Accrington Stanley right-back Nicky Hunt.