The Premier League strugglers have been searching for a new boss since sacking Brian McDermott earlier this month but, having held talks with Reading officials on Wednesday, Poyet has opted to stay put.
McDermott led Reading to the Championship title last year but he was fired on 12 March after a 2-1 home defeat to Aston Villa left Reading four points from safety, and on a losing run of four games in a row.
A 1-0 defeat at leaders Manchester United followed and the Royals are now seven points behind 17th-placed Aston Villa, with only goals scored keeping them off the bottom of the table.
They are reportedly keen to appoint a new boss before travelling to Arsenal a week on Saturday and are understood to have discussed the role with Poyet before he went on holiday on Wednesday.
But the 45-year-old Poyet, who has been with Brighton since November 2009, has decided to remain with the side who he led to the League One title and currently has in the thick of the battle for Championship play-off places.
Former Southampton manager Nigel Adkins is now seen as the leading candidate to replace McDermott, although caretaker boss Eamonn Dolan may yet be left in charge for the remainder of the campaign.