The 44-year-old, who led the McDiarmid Park side to an historic cup double in his first season as a manager, is quoted at odds of 16/1 in the latest betting markets – level with Frank Lampard and just behind Roy Keane (12/1).
Davidson had spells on the coaching staff of the Scotland national team as well as Stoke, Dunfermline, and Millwall before returning to Perth last summer to succeed Tommy Wright.
Celtic’s pursuit of former Bournemouth boss Eddie Howe collapsed on Friday, with a club statement citing “reasons outwith both his and Celtic’s control”.
The Hoops have confirmed their intention to unveil a new manager “very shortly”.
Former Australia national team boss Ange Postecoglou, currently manager of Japanese side Yokohama F. Marinos, is currently the favourite ahead of Scotland boss Steve Clarke, Hibs head coach Jack Ross, and ex-Borussia Dortmund boss Lucien Favre.