Dorrans joined Scottish Premiership club Rangers from Norwich City in the Championship in England in the summer transfer window.
The former West Bromwich Albion midfielder, who started his professional club football career at Livingston, is a Rangers fan.
Davidson, 29, has revealed what he told 30-year-old Dorrans in the summer.
“Even when he was in my youth team at Livingston, he was Rangers-daft,” Davidson told The Scottish Sun about Dorrans. “It’s not an act with him. He genuinely is a Rangers fan and was desperate to join.
“When I saw him in pre-season I had a laugh and joke with him and said ‘You finally made it’. I think having a couple of people like that in your team is a massive help.”
Rangers will return to action on Friday evening when they take on St Johnstone away from home at McDiarmid Park in Perth in the Scottish Premiership.
Pedro Caixinha’s side are third in the league table at the moment with 14 points from eight matches, six points behind leaders and Old Firm rivals Celtic.
Meanwhile, St Johnstone currently find themselves fourth in the standings with 14 points from eight games.