The manager is currently thinking through the last stages of his January transfer window plans and is believed to be set on a radical shake up of the squad.
The manager has been unable to look at either player since his arrival given the injuries which has sidelined them for weeks but is keen to build a younger and more athletic squad.
Moyes’ big winter requirement is a ball carrying midfielder but he is looking keenly at the back line and reinforcements could arrive to put Byram and Fonte’s future on the line.
With Pablo Zabaleta reaching his 33rd birthday on January 16 the manager knows that he has to bring in fresh blood at the back and he wants to take a good look at Byram too.
A Hammers insider explained: “The manager has made it clear he wants to bring in players and there could well be movement in both directions.”