The Sun claim Toffees boss Sam Allardyce is keen to add Premier League experience to his ranks and has up to £30million at his disposal to splash out on Evans.
A move back to the north west could suit the Northern Ireland international, who was on Manchester City's wanted list last summer.
The Baggies refused bids in the region of £25million from rival Premier League clubs last summer and were said to be holding out for £30million.
However, Everton may still find it impossible to sign the Northern Ireland international in January because new Baggies boss Alan Pardew is a big fan.
Pardew, who took charge of his first Albion match against Crystal Palace following the dismissal of Tony Pulis, tried to sign Evans when he was Newcastle manager.
The transfer failed to materialise and Evans stayed at Manchester United.
“There was a spell at Newcastle where I looked at Jonny for a while,” Pardew confirmed recently.
“He will certainly stay as captain.”