The 24-year-old was close to joining the Merseyside club this summer before Arsenal blocked a deal on deadline day following talks between the player and manager Mikel Arteta.
Maitland-Niles has become frustrated at a lack of first-team opportunities at the Emirates Stadium -- he took to Instagram near the end of the window to declare "all I wanna do is go where I'm wanted and where I'm gonna play" -- having spent the second half of last season on loan at West Brom, making 12 Premier League appearances.
Arteta told Maitland-Niles he remains part of his plans but sources have told ESPN that Everton are expected to test Arsenal's resolve to keep him January, particularly if he continues to struggle for opportunities.
Everton were previously in talks over an initial season-long loan with an option to buy before negotiations collapsed.
Sources have told ESPN they will try and secure a permanent move in January.
Maitland-Niles, whose contract expires in 2023, has regularly been used as a right-back or a wing-back but his preferred position is in central midfield.
Speaking ahead of that match on Thursday, Arteta reflected on Maitland-Niles' future after intervening to keep him at the club.
"Any private conversations with players, on the role we believe they can have in the team, is always discussed openly but privately and that is how we resolve the matters," he said. "I am happy he has stayed, that was our intention and he is another player in the squad."