Monk replaced Michael Laudrup at the helm in February and Saturday's 4-1 win over Aston Villa secured Swansea's Premier League status for next season.
Discussions are now set to take place with Monk to determine whether the 35-year-old will be handed the job on a long-term basis.
"Over the next couple of weeks, we are going to talk things through with Garry and reflect on his time in charge," said Jenkins in the South Wales Evening Post.
"The decision we are going to make will be a huge decision, and obviously it will be made in the best interests of the club going forward.
"Hopefully it will be a long-term decision that gets the club in the position where we want to be.
"We all agree that we would like to get everything sorted and clarified as soon as we can."
Speaking on Goals on Sunday, Monk confirmed he is planning for next season's Premier League as if he will be head coach.
"Everyone knows my affection for the club - I've been a player there for 10 years, now, and the responsibility on me I put on myself that when you step in you have to do something like this which, obviously, I haven't had the experience of but it's something I've been working towards," he said.
"To be fair, the chairman, the board and the fans and the players have been brilliant with me but I always put it on myself that the situation that we've been in - if it had gone bad, it would probably have been the worst feeling that anyone could have felt. But thankfully it has gone pretty much okay and we've got where we need to be.
"The truth is I haven't really worried about myself - I've just wanted the club and the players to get back to enjoying themselves and their football, and scoring goals.
"I said that once we are safe, that's when I'll worry about myself and not just that, but for the future for everyone - clarification for players, for fans, myself, the board.
"I have to prepare as if I'm taking the club forward for the future. I've done my preparation and it is still on-going; I have conversations with the chairman about possible targets and doing stuff but to be able to act on that stuff you need to be in charge.
"I've got a great relationship with Huw Jenkins and the board and I'm sure in the coming days that we'll have that conversation."