The England international is widely expected to pack his bags in the summer following more harsh criticism from Jose Mourinho.
And according to bookies Paddy Power, Goodison Park is currently his most likely destination.
Shaw is priced at 7/2 to join the Toffees where he could join former Man United forward Wayne Rooney.
Next up in the betting is a shock return to his former club Southampton at 6/1, followed by West Ham (13/2) and even Arsenal, (10/1).
Paddy Power spokesperson Amy Jones said: “It looks as though the Brighton game was the final straw for Shaw.
“There’s only so much mugging off one man can take, and we make Goodison Park his next club of choice. Though a return to Saints is an interesting prospect, too.”
Shaw has struggled to establish himself under Mourinho having been signed by Louis van Gaal for £30m from Southampton in 2014.