Everton's Phil Jagielka
Moyes made his first signing as Sunderland manager on Friday having landed Chelsea defender Papy Djilobodji to boost his back line, but more changes could be coming at the back.
Sunderland manager David Moyes
The Mirror report that Sunderland defender Lamine Kone has travelled to Merseyside in the last 24 hours ahead of an anticipated agreement with Everton, as the Ivorian prepares to leave just over six months after joining the Black Cats from Lorient.
It's added that Sunderland are keen on a part-exchange deal so they can land an immediate replacement for Kone, and Moyes wants centre back Phil Jagielka included in the transfer.
Everton's Phil Jagielka applauds the fans
Jagielka, who turns 34 next week, was signed by former Everton boss Moyes back in 2007 having starred with Sheffield United, and he's been a dependable play for the Toffees ever since.
With more than 300 appearances for Everton, Jagielka has been a regular at Goodison Park, and was appointed club captain by Moyes in 2013 after Phil Neville had announced his retirement.
Sunderland's Lamine Kone in action with Watford's Troy Deeney
Now, with Kone seemingly heading to Everton in an £18m deal, Jagielka could be heading in the opposite direction if Moyes gets his wish, as he's identified the England international as the man he wants to lead his defence next season and beyond.
Everton are allegedly set to land Kone and Swansea's Ashley Williams, and with Ramiro Funes Mori also in the ranks, Jagielka may be fourth choice under Ronald Koeman, making a reunion with Moyes and first-team football at Sunderland potentially appealing.
Would you like to see Jagielka arrive as Kone's replacement?