The 30-year-old, who spent last season on loan at Marseille, is back at Loftus Road; his reported wages of £70,000 a week have dissuaded the French club pursuing a permament switch.
Huyton-born Barton supported Everton as a boy and on Wednesday morning tweeted enigmatically: "Once a blue.."
Willie McKay told talkSPORT: "Obviously his package with QPR is pretty big for Marseille's terms really.
"Joey said there was only one club he would make a sacrifice for and it would be Everton and I made that call to Martinez.
"That is the club he loves. I have approached them and said Joey would love to have Everton on his CV.
"Fair play to him he said he would lose money to go to Everton - and not many professionals would do that.
"Roberto Martinez said he likes him as a player and he also spoke to (Martinez's assistant) Graeme Jones, who loves him as a player. Anything can happen in football. It could happen."
Everton are currently in San Francisco where they play against Juventus in the International Champions Cup on Wednesday night.