The long-serving defender and club captain, 35 next month, has not played since September after undergoing knee surgery.
Monk, who joined the club when they were in League Two in 2004, has long been viewed as a future member of the coaching staff.
He has already obtained the UEFA B licence and has been doing his A licence this season.
Although there has been no confirmation from the club, it is thought they are preparing to offer him a role on manager Michael Laudrup's backroom team.
There has also been speculation that Pep Clotet, a Spaniard brought in to run the academy last November, could step up after impressing the Swans hierarchy.
Reports of this possible coaching reshuffle come with Laudrup facing increasing pressure.
The Dane's side, who won the Capital One Cup last season, are facing a relegation fight after a loss of form.
The Welsh side have won just one of their last 10 Barclays Premier League matches and, although 12th, are only two points above the bottom three.
They face arch-rivals Cardiff in a crucial derby at the Liberty Stadium on Saturday.