Oldham enjoyed an impressive victory over Liverpool in the FA Cup little over a week ago, but Dickov left Boundary Park following a run of nine defeats in 10 League One games.
Despite still being under contract with Manchester United as a player, Scholes has been installed by bookmakers as a contender to replace Dickov, given his allegiances to Oldham.
"We initially made Scholes available to back as a sort of novelty option, but within 24 hours he's received the most interest," Sky Bet football trader Chris Spicer said.
"Oldham is a club known to be close to his heart, so if he was to receive a player-manager offer then perhaps he might find it too hard to turn down."
Scholes, 38, is a lifelong Oldham supporter and would undoubtedly be a popular choice as manager, although the Manchester United legend may lack the credentials to haul the Latics out of the relegation mire.
Along with Scholes, who is expected to retire as a Manchester United player for a second time at the end of the season, Iain Dowie and Phil Brown also have short odds to succeed Dickov.