The right-sided player looks destined to play higher up the pitch.
The Irish right-back picked up an injury in the recent Capital One Cup win against Manchester City and that sees Everton short in the right-back position.
That saw the rumour jump in terms of its popularity.
The Liverpool Echo picked up on the talk again this week after the injury to Coleman and Roberto Martinez admitting that the club could move for a new right-back.
So is Byram the man for this role?
That is not to say Everton shouldn't sign Byram. He is a top talent who has impressed in the Championship recently, especially in the past month or so.
But not as a right-back.
As he has matured into a first-team player it has become clear that he is a more advanced player and he has struggled to put in his best performances in the defensive position.
If Everton want a new right-back in they have plenty of options, Byram however should be thought of as a future attacking star and only a right-back at the last resort at Premier League level.