The Black Cats are currently three points adrift of safety in the table, though they do have two games in hand on the teams above them.
The North-East club plays one of those matches on Wednesday night when they face title challengers Liverpool at Anfield.
"Now we will see. We have got 10 games. I am not going to hide. I am going to be here and I am going to take responsibility, but let's wait and see," Poyet is quoted as saying by Sky Sports News.
"I am not going to be here just to have a job. I am here because I believe that we can be better than at least three teams. That's the aim from here to then."
Sunderland have failed to register a win in their last four Premier League matches.