Several reports from the UK and Ireland have indicated the former Lyon star is looking to leave the Toffees before the transfer deadline on September 1st.
O'Brien played the majority of last season out of position at right-back with the Republic of Ireland international hoping to be moved into his favoured centre-back spot for the 2026/27 campaign.
However, that does not appear to be in Moyes' immediate plans, after he remained on the bench for last weekend's Premier League season opening win over Crystal Palace.
Palace - one of O'Brien's former clubs have been linked with a possible offer - alongside Daniel Farke's Leeds United.
Moyes admitted he's confident of signing more players before the transfer window closes, including a new specialist right-back, but the Scottish boss indicated O'Brien is still in his thinking for the 2026/27 campaign.
"There's no problem there, I’ve just chosen at the moment not to use him, but he’ll be used in plenty of games, no doubt."