City sat just two points clear of the relegation zone when the club's hierarchy placed the 35-year-old in charge of first-team affairs on a temporary basis following Michael Laudrup's departure in February.
However, the Swans thumped Aston Villa on Saturday to open up a seven-point gap between themselves and the drop-zone with two games to spare, and the captain is convinced that Monk deserves a full-time position in South Wales.
"It's a massive relief for us and we are very happy with the performance and the win," the skipper told Sky Sports News following the win over the Villans.
"I said from the first week when I saw his work with us that he deserves the job. We are pleased that we may have stayed up.
"He deserves the job, we are all enjoying it, training is good so I think he really deserves his chance."
Swansea conclude their Premier League campaign by hosting Southampton at the Liberty Stadium next week, before travelling to Sunderland on the final day of the season.