The Las Palmas defender is also courted by Barcelona, but Everton are ahead of them at present.
The Uruguayan youngster has become hot property in Spain with a series of impressive performances catching the eye.
One of those interested are Barcelona, who are keen to add a young central defender to their squad for the future.
Indeed, the Catalan giants have already reportedly made their move for the player, offering €7million, around £5.5million, to secure his services.
That offer has been outright rejected by the club, who are keen to get as much money as possible for their young star.
A price of €10million, around £7.8million, has been touted, something Barca are not willing to pay, but Everton would be more than happy to offer according to Sport.
They see Lemos as a replacement for John Stones, who, according to the report, the Toffees are fully expecting to leave Goodison Park this summer.
Chelsea courted the Englishman, who is part of England’s squad for Euro 2016, last summer and a move to Manchester City has been mooted this year.
Should he leave, Lemos would represent a perfect replacement for him, and Everton's willingness to pay what Las Palmas want has put them at the front of the queue.