Garner is a product of the Manchester United academy and was expected to be given a chance to stake a claim for first-team football at Old Trafford this season.
However, Manchester United have decided to try and sell him this summer and there has been interest from several clubs in the Premier League.
Southampton have been regarded as the favourites to sign him, but Garner is also on the radar of Leicester and Tottenham.
And Everton are now claimed to be pushing forward with an effort to sign him and are now in discussions with Manchester United.
The Merseyside club have stepped up on their interest in Garner ahead of next week’s transfer deadline.
Manchester United want around £14m for a player who impressed at Nottingham Forest on loan last season.
Leicester are keeping tabs on his situation but their interest is dependent on outgoings at the King Power Stadium.
Garner was an unused substitute against Brighton and Brentford and was left out of the Manchester United squad against Liverpool.
