But Poyet has warned that the club require more than a change of manager to arrest their long-term decline.
Allardyce is reportedly among the candidates under consideration to succeed Dick Advocaat after the Dutchman resigned just seven fixtures into a disappointing start to the season.
Advocaat himself, however, only arrived at the club in March after Poyet had been sacked.
Since Roy Keane's departure in 2008 Sunderland have been consistently unstable, and Poyet, who impressed in leading them to Premier League survival in 2014 when relegation had appeared inevitable, is adamant that not only do they require the abilities Allardyce could provide but that their problems run significantly deeper.
"Now there is (another) manager leaving and it is a little bit too many," said Poyet, who was speaking at the Leaders Sport Business Summit.
"So I think it is clear now that it is not the manager. Sometimes when a team is not working you change the manager and things go well and you can say 'Good decision', but when it happens four or five times, come on, be realistic.
"I don't think you can blame Martin O'Neill, Paolo Di Canio, Gus Poyet and Dick Advocaat. There is something that is not working there. If I knew what it was I would call the chairman tomorrow but I don't. Fortunately it is not my job.
"But they have to look somewhere else. They need to find where to look. They need to take an appointment and stick with it whatever results come in the next two or three years.
"Sam Allardyce (should be Sunderland's next appointment). (But) first of all I don't know if Sam would like to go and maybe I'm putting it a little on Sam and he is going to kill me next time I see him."
The odds on Sam Allardyce becoming the new Sunderland manager have plummetted after a series of wagers.
Allardyce is the favourite to replace Dick Advocaat as head coach at the Stadium of Light.
The former West Ham boss – who played for Sunderland in the 1980s – is without a club and is the clear frontrunner for the post.
Bet365’s Steve Freeth said: “Punters have only been interested in backing Allardyce on Thursday with his odds plummeting from 7/4 into 1/5 in the space of a few hours.”
Latest odds (courtesy of bet365): Allardyce 1/5; Nigel Pearson 12/1; Bradley 20/1; Dyche 25/1; Moyes 25/1; Rodgers 33/1; Lennon 33/1; Hasselbaink 33/1.