Walker, 36, joined Burnley on a two-year contract from Manchester City last summer in a £5 million deal, after a proposed move to Everton fell through and following an earlier near-miss with the Merseyside club as a teenager.
The 96-cap England international made 35 Premier League starts for Burnley last season but could not prevent their relegation, and he opened this campaign by scoring the winning penalty in Saturday's Carabao Cup shootout victory over Notts County.
Should he complete the switch to Goodison Park, it would be for a nominal fee, giving manager David Moyes a low-cost, high-experience option to finally settle a position that has troubled Everton for several transfer windows, reports The Sun.
Everton's incoming business has already produced one headline moment away from the pitch. New signing Brennan Johnson delivered a pointed response to talkSPORT hosts Jamie O'Hara and Gabby Agbonlahor after joining the Toffees from Crystal Palace, in a swap deal that saw Dwight McNeil move the other way to Selhurst Park.
The exchange followed O'Hara's on-air description of the trade as swapping "c**p for c**p," a comment he later apologised for alongside Agbonlahor.
Johnson responded on Instagram with a simple message aimed at both pundits: "Won more than both of these put together."
The claim holds up: Johnson is a Europa League winner from his time at Tottenham and a Europa Conference League winner with Palace, and also won the Championship play-off final with Nottingham Forest earlier in his career. O'Hara's only major honour is a League Cup with Spurs in 2008, while Agbonlahor never won a trophy in his playing days.
The row began when Agbonlahor compared the two players' underwhelming statistics from last season, prompting O'Hara's crude summary of the trade, which the pair then laughed about on air. O'Hara later said the comments had been taken out of context, insisting on social media that he had been discussing the pair's poor form last season rather than dismissing them as players, while Agbonlahor issued a separate apology, saying he should have "stuck up" for Johnson rather than laughing along.
Johnson, 25, has signed a four-year contract at Everton just seven months after arriving at Palace for a then-club-record £35 million fee from Spurs, a switch that yielded a Conference League winner's medal but no goals and just two assists in 26 appearances for the Eagles.