Moyes offered him a contract but Iwelumo had already decided on a move to Stoke.
Iwelumo, 34, explained: "His brother Kenny was my agent when I was with Aarhus in Denmark and I had a week on a trial at Preston and it all went really well.
"I played in a reserve game against Port Vale and John Rudge, who works for Stoke, watched me and offered me a contract with them the next morning.
"Preston offered me a contract that night but I had already signed for Stoke earlier that day. So I have Moyes to thank for bringing me back to England but Rudge jumped on the bandwagon!
"I didn't know the Preston offer was coming until after I had signed for Stoke. I had a great week working with David, who is a fantastic manager and has brought Everton on in leaps and bounds in his time there.
"He has a job for life there if he wants it and is being touted for all the big jobs like Manchester United, which is deserved.
"On the budget he has at Everton, compared with other clubs, he really has worked wonders at Goodison Park."