Poyet's team, who are seven points adrift of safety, face Manchester City at the Etihad Stadium on Wednesday evening knowing a sixth successive defeat will push them ever-closer to the drop.
And Poyet has cast an envious eye along the road to Newcastle, where counterpart Alan Pardew finds himself entrenched in a battle to win over his club's fans despite the Magpies sitting ninth in the table.
Asked to characterise the challenge of managing in the north east, Poyet said: "The most important part, which is the fans, they love it, they love football and they are desperate to do well.
"Go across the river and go to Newcastle - they are in an unbelievable situation in the table and there is a nightmare.
"Can we change over? Because that is a great problem to have, to be ninth?
"Why? Because it's that drive to get better all the time and to be always on top and it doesn't matter how many games you have won in the past because you need to win the next.
"That part of football, I love it because it's the best way to do it, to try to get better all the time.
"In terms of mentality for me, it's perfect, the north-east. It's always better if you finish ninth instead of 10th, and I am like that.
"The problem is at the moment it is so bad that it's not enjoyable, so the idea is to make sure that we don't put ourselves in this situation anymore.
"It doesn't matter who is the manager, I am talking about the club now, the club itself."
Poyet is convinced Sunderland's problems are rooted deep inside the club and not just in the identity of the man who occupies the manager's office, and recent history suggests he may have a point.
The 46-year-old Uruguayan is the sixth man to do the job in less than six years, and he believes that tells its own story.
Poyet, who admitted he was unaware of former chairman and manager Niall Quinn's past reference to the presence of "gremlins" on Wearside, said: "It's true that from outside it has been always, looking at this football club and its fans, a very difficult place to come.
"But everybody keeps talking about when Peter Reid was here with Niall Quinn and Kevin Phillips, and after that I don't think it was a difficult place to come.
"We could be talking for two hours about what we think and everything goes back to the same place, so I don't think those two years can put any expectations in the minds of fans who have been here for 50 years or the kids or the new fans who have been here maybe 10.
"I think sometimes when things don't work, it's natural that you change the manager and get another and if it doesn't work again, you have got a chance to change the manager again.
"And then the third time, I'd like to think about it because I don't think it's the manager anymore when it's four or five.
"Do I try to get away from my responsibility? Not at all. I'm in charge, I pick the team, I am responsible.
"Now if you want me to lie, I have got no problem. It's very easy, it's easy to lie. I can tell you any lie you want to hear and agree with everybody, and this club is going to be in the same situation next year with me or with somebody else."