The Rangers midfielder sits just ahead of three other team-mates: Ryan Jack, James Tavernier and Connor Goldson.
All four have attempted over 500 passes so far this campaign with Davis topping the list at 585.
The only other player to break that mark is Celtic midfielder Callum McGregor, who sits ahead of team-mates Boli Bolingoli and Scott Brown in sixth and seventh respectively.
It thus far represents a marked difference from recent seasons where Celtic have dominated the top of the leader board.
McGregor was first last season ahead of Brown, who finished top of the pile in each of the previous two campaigns.
Brendan Rodgers’ first year as manager, 2016/17, saw Celtic occupy five of the top six positions in the passing table.
At present, Rangers lead their rivals in terms of average possession, which sits at 70.7 per cent. Celtic are currently second with 61.3 per cent.
Celtic finished at least nine percent higher than Rangers in each of the previous three seasons since the Ibrox club’s promotion to the top flight.