Monk has urged Swansea's owners to clarify whether he will still be at the helm next season as soon as possible having secured the club's Premier League status last weekend.
And Huw Jenkins has indicated that while no decision is likely this week, Monk's fate should be settled before Swansea's season comes to an end at Sunderland in 12 days' time.
"Over the next couple of weeks we are going to talk things through with Garry and reflect on his time in charge," said the Swansea chairman.
"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."
Head coach Monk has made it clear that he wants to be named Swansea boss on a permanent basis, having originally been appointed for the "foreseeable future" following Michael Laudrup's sacking in February.
And though it may take a little while to come to a decision, Jenkins admits it will be best for all concerned to get the managerial position sorted.
"We all agree that we would like to get everything sorted and clarified as soon as we can," he added.
A string of Swansea players have come out in support of Monk, who has taken 15 points from 12 league games at the helm to ensure the club will be in the top flight for a fourth successive season.