The pair have been in touch for weeks discussing plans to collaborate once the competition was over, according to The Sun.
He reportedly told Arthur, who beat Jahmene Douglas to be crowned champion last night: "Well done. Let's get into the studio together as soon as the show's done."
A friend of Sheeran said: "Ed sees some of himself in James. He likes his style and is convinced a collaboration would be a big hit."
Arthur has also received offers to write or collaborate with a host of other performers, including Rita Ora and Labrinth, who is signed to Simon Cowell's Syco.
He has said that he wants to release his debut album "as quickly as possible". His first single 'Impossible' - originally released by Shontelle - is currently number one on iTunes.
Photo gallery - X Factor final 2012:
Copyright: Ken McKay/Thames/Rex Features