Keane is about to break up with Burnley at the end of the season.
Manchester United signed Ivory Coast global Eric Bailly in a £30m deal from La Liga outfit Villarreal in last summer's transfer window.
Keane's former club Manchester United are due 25 per cent of any transfer fee.
Keane made his Burnley début against Leicester City.
Keane, who made the permanent switch to Turf Moor from Manchester United in 2015, has enjoyed a meteoric rise in the Premier League this season and a host of clubs are vying for the defender.
Keane, 24, has been linked with a return to his first club Manchester United, though we understand no discussions have taken place between United and the player's representatives.
Everton manager Ronald Koeman publicly admired the defender in January, although Keane is said not to be on Tottenham's radar as source confirms.
It was more recently reported that Dier, who enjoyed a sensational breakthrough campaign under Mauricio Pochettino last season, was also on his radar.
It is peculiar to see Mourinho seek out Dier as a replacement for Carrick at this given time.
With Burnley virtually safe in the Premier League ahead of Sunday's home game against United there will be no upset from supporters about the Keane news breaking now, especially if it fires the starting gun on a bidding war for the player's services.
'At Southampton previous year we finished with 63 points.
The Dutchman hailed their season as "incredible" although is mindful that form away from Goodison Park must improve after only two wins on the road since September.