David Moyes is still working to bring in reinforcements in the coming days, but reports from Football Insider claim O'Brien wants to join another Premier League team, amid frustration on Merseyside.
O'Brien played the majority of last season out of position at right-back with the Republic of Ireland star hoping to be moved into his favoured centre-back spot for the 2026/27 campaign.
He was an unused substitute for Everton's Premier League season opening win over Crystal Palace and Moyes confirmed the club are chasing several right-back targets.
However, Moyes appears to favour Michael Keane, James Tarkowski and Jarrad Branthwaite over O'Brien, and he could move on if right-back cover is secured.
Former club Crystal Palace are interested in the 25-year-old - following the departure of Maxcence Lacroix - and they could have the edge over Daniel Farke's Leeds United.
With O'Brien open to moving on and his contract running until 2028, Everton will demand around £30M for a sale.