The 22-year-old is out of contract at the end of this season and the Toffees have agreed a fee which is likely to be worth up to Â£5million.
Everton will now discuss personal terms with Byram, who has been extensively scouted by a number of Premier League clubs.
Byram is a long-term option which could provide a short-term fix as manager Roberto Martinez is currently short of cover at right-back.
Republic of Ireland international Seamus Coleman is sidelined for another three weeks with a calf strain sustained in last week's Capital One Cup semi-final first-leg victory over Manchester City, while Tyias Browning has been ruled out for the season with a knee injury and veteran Tony Hibbert's fitness issues mean he has not played a match in 13 months.
John Stones, who has established himself as first choice in the centre of defence alongside captain Phil Jagielka, filled the void in Wednesday's goalless draw at Manchester City but the arrival of Byram, a Leeds academy graduate who has turned down the offer of a new contract, would give Martinez some much-needed back-up.
Source : PA