Bryan, 24, operates as a left-back for The Robins and is set to drawn attention from numerous Premier League clubs this January, with Newcastle United also apparently interested in a making a move for the defender.
With a defender - specifically a left-sided utility man - high up the list of targets, Bryan falls under the umbrella of what the Blues require during the upcoming window.
But the former England boss wants to make his first signing ‘the right one’ and may opt to target a forward before solidifying the back line.
Everton are in the market for a left-back with Leighton Baines still sidelined and only Cuco Martina - a natural right-back - their other senior option on the left-hand side.
Luke Garbutt was not registered for Premier League duty by former boss Ronald Koeman and has been playing out the season with Everton’s under-23s.
A regular on the scouting list
According to the reports, Everton and Newcastle have been regulars at Ashton Gate, keeping a keen eye on the emerging left-back.
Bryan has progressed through the ranks at Bristol City since signing as a schoolboy and has made over 200 first-team appearances since making his debut in 2012.
With the Robins flying high in the Championship table, sticking with his boyhood club and perhaps making the jump to the Premier League may be an option for the 24-year-old instead of making the move away.
It would also likely take a substantial fee to persuade City into selling one of their key assets and perhaps spurning a chance at promotion.