The Dutchman has made nine appearances since arriving at the Liberty Stadium for a second temporary spell in January, and scored as the men from South Wales closed their season with a 3-1 Premier League win at Sunderland on Sunday.
Emnes has regularly been linked with a transfer to Swansea since his first stint at the club in 2010, but Monk - appointed as former boss Michael Laudrup's long-term successor on Wednesday - hinted a permanent deal could now be on the horizon.
"I've had a chat with Marvin and obviously we'll discuss it with him," he said. "Marvin knows how I feel about him and what I feel for the future, and we'll have to let the club speak to his representatives.
"We'll go from there."
But Monk insisted the club would be doing all they could to hang on to prized asset Wilfried Bony, who scored his 25th goal in all competitions in the win at Sunderland.
"We're lucky we're in a position where we don't have to sell anyone," he added. "It's up to us.
"Bony's one that we'll be keeping, he's one that we want here.
"He's one that has improved massively with us and, for me, I think there's an even better player than what we've seen this season from him.
"Him and a number of others I think you'll see are better players next season."
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