England centre-back Stones has been heavily linked with a move to Stamford Bridge, with Chelsea said to have made three approaches for the 21-year-old.
Chelsea's most recent bid was thought to be in the region of $65million (AUD) as the Premier League champions aim to bolster their resources at the heart of a defence that has struggled so far this season, shipping seven goals in three league games.
Stones reportedly submitted the transfer request on Tuesday before training ahead of Wednesday's League Cup clash with Barnsley, whom he left to join the Merseyside club in January 2013.
Everton officials have gone to ground while manager Roberto Martinez was coy when pressed on the issue of Stones' future in his pre-match press conference.
"There is nothing to announce and nothing to discuss. There's nothing to comment on that," he said.
Since making the move to Goodison Park Stones has made 57 appearances for Everton, scoring once while also earning four England caps.