The 32-year-old has spent seven years at the Liberty Stadium and Swans boss Brendan Rodgers is keen to tie his skipper to the club for the remainder of his career.
"Garry deserves a new contract," Rodgers told the club's official website.
"He is a wonderful captain - a great leader for us both on and off the field.
"There's no doubt that over the next few years we will have to look after him and manage him - that's something you have to do with a lot of players who are over 30."
Monk could miss the start of the Premier League season with a foot injury.