The 36-year-old joined the Clarets in a £5M deal last summer after falling out of favour at Manchester City on the back of an incredible eight years at the Etihad Stadium.
Walker played in 36 of Burnley's 38 Premier League games in 2025/26, but he was unable to prevent their relegation from the top tier of English football, and he wants to keep playing at the highest level if possible.
Burnley could listen to offers on him, as he remains under contract until 2027, and Everton are rumoured to be looking at him as an option to fill their vacant right-back spot.
The departure of former club captain Seamus Coleman in June means David Moyes is short on both experience and cover in the role and Walker could fit the brief he needs.
The Toffees are exploring a list of options, including Celtic's Alistair Johnston and West Ham United defender Aaron Wan-Bissaka - plus England U21 international Brooke Norton-Cuffy at Genoa - but Walker offers over 400 games at Premier League level.