Earlier this week it was believed that the 34-year old former England and Liverpool striker had rejected an offer from an Australian club following his release by Aston Villa but more recently it has come to light that the Jets are closing in on his signature.
It is understood that Heskey, who has also played for Leicester City, Wigan Athletic and Birmingham City during his career, will have a very meticulous medical as he continues to battle with ongoing injuries.
Jets head coach Gary van Egmond indicated that the club were indeed interested in Heskey and said a deal could be finalised in the next couple of days.
"We've just seen Alessandro Del Piero sign for Sydney at 37," said van Egmond.
"If you are looking at someone at 34, there is no issue with age. It doesn't matter about age so much, it's more about how sound they are and where their body is – that's the most important thing.
"You are really excited if you can bring someone like Emile Heskey to your football club, someone with that stature and quality.
"We are looking for a player to play upfront. We've always been looking for a number 9 and it's fallen through a couple of times.”
He added: "Usually if things do happen, they happen pretty swiftly. I would imagine that it will be one or two days maximum until we get an outcome.
"In that respect, I would suggest that's what could occur. If you see something drag on, from my experience, it will never eventuate."