The Magpies could still scupper the 22-year-old's proposed move to Goodison Park.
The 22-year-old was given permission to discuss personal terms with Roberto Martinez's side, after they had their bid of £3.7m plus add-ons accepted by the Yorkshire side (via The Guardian).
However, The Guardian claims that the Magpies - along with West Ham - are still interested in bagging Byram's signature themselves, and have made an approach in an attempt to bring him to St James' Park.
Despite their current position in the Premier League's relegation zone, a move to Newcastle may well appeal to Byram. Steve McClaren's side have signalled their intent to pull themselves out of the trouble they find themselves in by signing both Henri Saivet and Jonjo Shelvey.
And with the club boasting as much potential as it does - due to its stature - Byram may relish the prospect of being a part of the side that may possibly begin to take Newcastle to the level they feel they should be.