The 30-year-old centre-back has been a peripheral figure of the squad this season and has made just one substitute appearance in the Premier League.
Frank Lampard favoured Conor Coady and James Tarkowski as the centre-back pairing and Keane was reduced to being a constant presence on the bench.
The defender has been tipped to leave Everton, with Southampton holding an interest in taking him to the south coast to aid their own battle against relegation.
Everton are also set to appoint Sean Dyche as their new manager today and it has been claimed that Keane is now set to stay.
The defender enjoyed success with Dyche at Burnley and the boss played a big role in his development.
With the former Burnley boss set to take charge of the club, the centre-back is now expected to continue at Everton.
It remains to be seen whether he goes straight back into the team under Dyche ahead of their game against Arsenal a week from Saturday.
