The 35-year-old, who has been at the Welsh club for 10 years, took charge of the first team in February following the departure of Michael Laudrup.
Monk has revealed that he has held talks with chairman Huw Jenkins, and is optimistic that he will be able to lead the Swans next season.
"We'll see," Monk told reporters in Thursday's press conference. "Of course I am hopeful. It is something I really enjoy doing, I've learnt a hell of a lot in a short space of time, and once you get a taste of it you don't want to let it go."
Swansea, who are currently 12th in the Premier League table, have two games left to play this season.