The Premier League's joint-bottom side are seeking a replacement for Brian McDermott, who was sacked earlier this month after a run of four straight defeats.
Neither club has commented on the speculation but Poyet has been backed into odds of 1-10 with several bookmakers.
The former Uruguay international took the Brighton reins in November 2009 and in his first full season in charge led the club to promotion as League One champions.
He is contracted to the south-coast club - currently outside the play-off places only on goal difference - until 2016.
Nigel Adkins started the day at the head of the market but has now drifted out to 11/2, while caretaker boss Eamonn Dolan and Bradford manager Phil Parkinson are both 12/1, with initial favourite Paolo Di Canio now 25/1.
Sky Bet's Chris Spicer said: "The money has poured in across the industry for Poyet to take the job, promoting us to significantly cut his price.
"While a Premier League job initially appears an ideal step up for the highly-rated Brighton boss, it might be considered a questionable move with the Royals 1/16 for relegation, while his current club are 9/2 to reach the top flight as they sit just outside the Championship play-off places."
Next Reading manager odds with Sky Bet:
Gus Poyet 1/7
Nigel Adkins 11/2
Phil Parkinson 12/1
Eamonn Dolan 12/1
Luis Enrique 18/1
Paolo Di Canio 25/1
Odds subject to change.
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