Hart has been dropped by new City boss Pep Guardiola, and is reportedly considering his options amid the expected arrival of Barcelona goalkeeper Claudio Bravo.
Everton, meanwhile, allowed Tim Howard to return to his native United States earlier this summer and are thought to be seeking another addition between the sticks, despite landing Maarten Stekelenburg from Fulham last month.
Speaking to Talksport on Wednesday, Barton, who grew up a Toffees supporter, said he expects Hart to make the move to Goodison Park before the transfer window closes in two weeks' time.
Asked by fellow Liverpudlian Kevin Nolan if he would like to see Hart at either Merseyside club, the 33-year-old replied: "Yeah. I think he’ll end up at Everton - even though Liverpool probably need him as well.
"I think the way Everton are going at the minute - shelling it out - it’s burning a hole in their pocket. I wouldn’t be surprised to see Everton go and offer £50 million or something stupid for him."
The arrival of British-Iranian billionaire Farhad Moshiri, not to mention this month's £50 million sale of John Stones to City, has seen Everton's power in the transfer market increase.
And Ronald Koeman's side are finally starting to leave their mark, bringing in Aston Villa midfielder Idrissa Gueye, Swansea City defender Ashley Williams and Crystal Palace winger Yannick Bolasie for a combined £44 million this month.