The Royals are looking for a new manager after sacking Paul Clement on Thursday following their dismal Championship form that has left them in the relegation zone.
Ex-Aston Villa boss Bruce is the bookies front-runner at 3/1 after a spike in bets.
Bookmakers have seen 54 per cent of all bets backing Bruce to be the next man in charge of the Berkshire-based club.
Bruce, 57, has been linked with the job in the past when Clement was under pressure in October.
Bolton boss Phil Parkinson is the second favourite for the Reading job at 5/1.
Oddschecker spokesperson George Elek said: “Aston Villa’s form has turned for the better since Bruce left the club, so the former Hull boss will be desperate to get back into a job to prove that he is still a top manager at Championship level.
“He would at least provide a safe pair of hands, which is just what Reading need to ensure safety.”
Bruce has been out of work since being relieved of his duties at Villa Park in October – Dean Smith succeeding him.
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