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 in to odds-on favourite 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.
Reading striker Jason Roberts can understand why Poyet, who is under contract at Brighton until 2016, might be a target for his club's board.
He told Sky Sports News: "Gus Poyet's a good manager, he's done very well at Brighton so we'll have to wait and see what happens.
"Whoever comes in, we've got a good bunch of guys with the right motivation to achieve things in the future."
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.
After seeing the new favourite cut to 1/7 from 7/2 this morning, 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."