The Dutchman moved to Wales on loan in January - his second such spell at the club - and scored once as Monk steered the Swans to Premier League safety.
Now Monk has landed the Swansea job on a permanent basis, he admits he would like to include Emnes - who has a year to run on his contract in Teesside - in his future plans, with the player also open to a move.
Monk told the South Wales Evening Post: "I've talked to Marvin about how I feel going forward.
"I like him, but he needs to improve on a lot of things - his fitness for one.
"It is a case of the club speaking to his representative and seeing what happens.
"There's no definites here and there just yet but we will see in the coming weeks."