Poyet's men succumbed to a 1-0 home defeat to Everton on Saturday, a Wes Brown own goal securing a result that keeps bottom-placed Sunderland seven points adrift of safety with six games remaining.
The Uruguayan has guided Sunderland to a League Cup final this season, but has won just six league games since taking over in October last year.
Predecessors Steve Bruce, Martin O'Neill and Paolo Di Canio have all been dismissed from their posts in the last five years after failing to produce consistent results with Sunderland.
And with the Wearside club seemingly heading towards the drop following a dreadful season, Poyet believes the issues at the Stadium of Light extend beyond performances on the pitch.
"I think there's something wrong in the football club and it's not an excuse. I need to find that. If I don't find it, we've got a problem," Poyet said in quotes reported by the British media.
"I think I know what it is but no, I don't. You know, you think. It's too many times, too many things. I always say to myself 'What happened with Brucie (Bruce) in the second year?', 'What happened with Martin O'Neill?', 'What happened with Di Canio?' and what happens with me now?'
“I don't want to get away from the responsibility because I said on Monday night I am responsible. I am the first one.
"But who is going to be next? A, B, C - you can call him anything and the club will be in the same situation. I don't like it, it's not me. I need to find the solution and whatever it is, I need to put it there."
