Evans confirmed that Byram has been given permission to speak to a Premier League club about a move by Leeds president Massimo Cellino and talks are already under way with Everton, according to Sky sources.
Everton manager Roberto Martinez said his club had made an approach on Friday, although he suggested there was still some work to do before a transfer could take place.
And Evans has revealed other Premier League clubs are interested in Byram.
The Guardian claimed on Saturday that West Ham and Newcastle were also competing for the signature of the 22-year-old, who is out of contract at Elland Road at the end of the season.