The veteran midfielder is on the transfer list after slipping down the pecking order on Wearside.
First Division outfit Millwall offered him an escape route prior to the deadline at the end of August.
But the Swede's agent Vincenzo Morabito has revealed Schwarz spurned the chance to join The Lions.
"Millwall approached Stefan, they were looking for a charismatic midfielder," confirmed Morabito.
"But he said no, he didn't want to go there."
Sunderland are eager to offload the former Arsenal schemer from the wage bill and are believed to be prepared to still pay a percentage of his salary if he can secure a move when the transfer window reopens in January.
Black Cats boss Peter Reid has informed Schwarz he is out of the first team picture and will be anxious for a club to snap up the £4 million signing in the New Year.
Meanwhile, Sunderland's young goalkeeper Michael Ingham has had his loan spell at Stockport extended by a further month.
Ingham has been at Edgeley Park since the end of August but has made just one first team appearance for the Second Division outfit.
But with his opportunities in the North East limited, Reid has been happy to permit Ingham to stay with The Hatters to garner greater experience.
