The 33-year-old centre-back made 18 appearances for the Swans last season, however he has been told by new manager Michael Laudrup that he is not likely to feature regularly this term.
"Garry is a special case because he has been here for so many years and has a lot of experience," said Laudrup. "I spoke to him a couple of weeks ago and said it will be difficult for him to play in the starting XI. He is now making up his mind.
"I think he will make the right decision for himself whatever that is – either staying here or going somewhere else for the next couple of seasons.
"I always say it is easy to smile and shake hands for the picture when people arrive. But it's nice to shake hands when they leave and that doesn't happen often in this world of football."
Bristol City are reportedly very keen to sign Monk if he decides to end his eight-year stay with the South Wales club.