Madrid are asking for at least €25 million for the Colombia international, while Everton want to pay less, believing a lower fee would be appropriate given the state of the transfer market.
Nonetheless, sources have told ESPN that Real are optimistic about their chances of reaching an agreement soon.
James was given permission to stay away as players returned for preseason training at Valdebebas on Monday as he prepares for a move to the Premier League.
The connection between the playmaker and Everton boss Carlo Ancelotti, with the pair having worked together at the Bernabeu in the 2014-15 season and again at Bayern Munich two years later, has been key to getting a deal done.
Ancelotti tried to sign James when he was in charge of Napoli in the summer of 2019, but the player decided against the move, preferring to stay in Madrid.
He was keen at that time to join La Liga rivals Atletico, a transfer that was blocked by Florentino Perez after Diego Simeone's team won a preseason Madrid derby friendly 7-3.
A poor season followed, with James making just five league starts as he struggled to convince coach Zinedine Zidane.
His campaign reached a low when he was left out of the squad entirely for a La Liga match with Athletic Bilbao in July. Zidane said "James wanted to be left out for his own reasons" when asked about the player's absence.
"I want to go where I can play, where I'm happy," James said. "[Zidane] has his preferences for some players and that is respectable. When you see you don't get the same opportunities as your teammates it is difficult... [this season] is one of the biggest disappointments of my career."
The breakdown in that relationship, combined with Madrid's desire to bring in funds during this transfer window, made a departure likely this summer.
Until recently the club had received no firm offer, but Everton's interest has changed that, and a deal is expected to be completed shortly.