Talks over fresh terms have so far failed to reach a resolution, though the Toffees have not given up hope of keeping their combative midfielder beyond the summer.
Gueye's previous deal contained a clause to automatically trigger a 12-month extension, but that required a set number of Premier League starts which the 36-year-old did not reach last season.
He turns 37 in September, though David Moyes still sees a role for the Senegal international as he reshapes his midfield options.
Everton have already brought in Hayden Hackney from Middlesbrough for a fee of around £16m, but Moyes wants further reinforcements in the middle of the park.
Parma midfielder Mandela Keita has been offered to Everton, though the club have yet to move for the 22-year-old.
With no fresh proposal yet agreed, David Moyes's side have stopped short of confirming Gueye's departure, leaving a slim chance a new offer could still convince him to stay on Merseyside.
Gueye has made more than 200 Premier League appearances across two spells at Everton, having first joined in 2016 before returning in 2022 following a stint at Paris Saint-Germain.
He was a regular starter in central midfield last season alongside James Garner, and has been away on international duty with Senegal at this summer's World Cup.
Contract discussions were put on hold while the tournament played out, with Senegal since eliminated, and the two parties are expected to reopen talks now Gueye is back from international duty.
The uncertainty comes during a busy summer of transfer business at Goodison Park. Everton missed out on European qualification last term, their first campaign at the new Hill Dickinson Stadium, and Moyes is keen to strengthen across the pitch before pre-season begins.
Club captain Seamus Coleman has already departed following the expiry of his own contract, while a permanent deal to sign Tyrique George from Chelsea is also understood to be close to completion.
Should Gueye leave, it would end a memorable Everton career spanning two spells and more than a decade on Merseyside in total, having established himself as one of Moyes's most trusted midfield operators.