Monk was appointed as head coach "for the foreseeable future" after Swansea parted company with Michael Laudrup three months ago.
The 35-year-old revealed this week that he had recently had talks with Swansea chairman Huw Jenkins, while the club's board of directors were understood to be discussing the situation on Friday.
It is not thought there will be any announcement ahead of Swansea's Barclays Premier League clash against Liberty Stadium visitors Southampton on Saturday.
Swansea secured their top-flight survival with a 4-1 victory over Aston Villa last weekend, but they had flirted with the danger zone for a number of weeks.
"I think he (Jenkins) is happy with the work I've done," Monk said.
"I hope it's a case of everyone saying they are happy. You would have to speak to them.
"It will be what it will be. I can't affect what other people will say or decide in that room.
"I've done what I've had to do to the best of my ability and given my views. That is all I can do."
Monk faced a tough task after Laudrup left the Welsh club during what has been their most testing season since securing Premier League status.
But although there have been setbacks in terms of results along the way, he has ultimately masterminded a position comfortably away from trouble.
"Going forward, there will be hard times again, but I don't think I will ever be in a harder situation. It will never be as hard as this," Monk added.
"If I can show what I can do in this situation with limited experience, it bodes well for me in the future. That's what I feel, and hopefully others agree. I've really enjoyed it, loved every moment and learned a lot.
"It (meeting with Jenkins) was just a review of how things have gone. The whole season, what we could have done better and what we have done well.
"It was about what the future holds in terms of what I think and in terms of what the club thinks from Huw's perspective. That is the right way to do it, that is the right process for the club and that is what I expected, to be honest. I had no problems with that.
"I have asked the players to work in a different way, and taking that into account I think I have done the job well enough.
"There are things I need to improve on, but it is the same as anything, and I am sure we could say that about all our jobs.
"The bottom line is I have enjoyed it and I have done what has been asked of me."