The striker is reported to have informed his current employers of his desire to move on this summer, having spent the last two years in Germany.
The Russia international has had his head turned by interest from the Premier League, with Liverpool and Tottenham also believed to be keen.
His representative has done his best to curb the rumours, claiming that no negotiations have taken place.
He has, however, confirmed that Sunderland are among those to have held informal discussions regarding the 27-year-old frontman.
"They are one of the clubs who approached us concerning Pavel," Oleg Artemov told Sovetsky Sport.
"Besides them, there are some other teams from England and Germany who contact me from time to time, testing the waters for possible cooperation.
"So we just call each other and talk. Nothing of that went as far as some negotiations between the clubs.
"Pogrebnyak has proved in Germany that he is a skilled football player.
"It would be interesting for me to watch him play in the English league, but it's up to the player.
"One thing I can tell you for sure is that Pavel will not be without a decent team."
Artemov claims there is no guarantee that Pogrebnyak will be on the move this summer, with it possible that he could see out his contract at Stuttgart and walk away on a free transfer in 12 months' time.
He added: "Nothing has changed. Pavel has one more year left on his contract. Usually in this case the club would begin talks about a renewal or a sale, but Stuttgart keep silent.
"I can imagine a situation where Pavel sticks with Stuttgart until the end of his contract and moves to another team as a free agent. Any option is possible."