The 30-year-old spent last season on loan at Marseille and has made no secret of his desire to leave QPR following the club’s relegation from the Premier League.
A return to the south of France to re-join Marseille on a permanent basis has been mooted in the media, but Willie McKay, Barton's representative, has now revealed he would be keen on a move to Goodison Park.
According to McKay, Barton would take a pay cut to link up with the club he supported as a boy growing up on Merseyside.
"I spoke to Roberto to see if he would be interested in taking Joey," McKay told The Daily Mail.
"I told him that Joey would love to join Everton and that they are the only club he is willing to lose money to play for."
Barton's often controversial career has also so far included spells at Manchester City and Newcastle United.
Martinez has been particularly active in the transfer window since being announced as David Moyes' successor, signing Arouna Kone and Antolin Alcaraz from his former club Wigan Athletic, while Gerard Deulofeu has arrived from Barcelona on a season-long loan.
