The 48-year-old actor, better known as Minty Peterson on the Walford-based soap, allegedly started dating the BBC production assistant three months ago after they met on the set of Let's Dance For Comic Relief.
Parisi has now moved Wyer into his home in Highgate, North London and plans to marry her later this year, The Sun claims.
A source revealed: "They are so happy - Cliff wasted no time in getting down on one knee to pop the question."
Parisi teamed up with his soap co-star John Partridge (Christian Clark) for Let's Dance For Comic Relief in February, but the pair failed to make it through to the charity show's final.
The news comes just one day after Walford's Jo Joyner (Tanya Branning) announced that she is pregnant with twins.