Lamine Kone was on Merseyside on Tuesday with discussions continuing over a move to Everton, according to Sky sources.
We also understand that the Toffees placed a bid of £18m for the Sunderland defender more than a week ago, although David Moyes' new club have denied receiving any offer.
Mark McKay, Kone's agent, has told Sky Sports News HQ: "Lamine will train with Sunderland on Wednesday morning and is available for selection for Saturday's Premier League opener against Man City.
"In the meantime Lamine still awaits the offer of a new contract from the club."
Moyes, appointed at the Stadium of Light in July, is keen to bring in reinforcements for the start of the new Premier League campaign and has recently signed defender Papy Djilobodji from Chelsea.
There have been four new arrivals at Goodison Park so far this summer with the acquisition of Maarten Stekelenburg, Bassala Sambou, Chris Renshaw and Idrissa Gueye.
Kone, born in the Ivory Coast, was signed by Sunderland in January for £6m from French club Lorient.
He made 15 appearances for Sunderland under their former manager Sam Allardyce in last season's Premier League, which even included him scoring twice in their penultimate match at home to Everton.