Bayern Munich forward Toni Kroos left the door open for a switch to the Barclays Premier League in the summer.
Tim Sherwood claims the "silence is deafening" from the Tottenham board amid speculation he could be replaced as manager.
Poyet, assistant manager Mauricio Taricco and first-team coach Charlie Oatway have all been told to stay away pending an internal investigation.
The Uruguayan is believed to have had a series of disagreements with the club's board over his transfer budget for the summer.
But Brighton declined to reveal the reasons behind the suspension of Poyet, who has three years remaining on his current contract.
A club statement read: "Brighton & Hove Albion has confirmed that it has suspended its manager, assistant manager, and its first team coach from their duties.
"The club will now follow its internal procedures with regard to this matter and will make no further comment at this time."
Poyet's future was already the subject of intense speculation linking him to a number of vacancies.