The 35-year old will reportedly put pen to a 12-month deal worth $750,000 with the A-League club despite interest from Russia, the Middle East and Asia.
A Jets source told AAP: "He's staying. The money's right and the deal's been done.
"It's just a matter of fixing up a few of the minor details which will be a formality.
"Heskey is planning on going home to England at the end of this season for three months before coming back out in August for the start of pre-season training.
"One of the things he'll be sorting out when he goes back home is his plans for his family.
"They've stayed in England while he's been out here but bringing them back out with him now that he has sorted things out is obviously something he will look at."
Since arriving in Newcastle from Aston Villa in September, the former Liverpool, Leicester City, Wigan Athletic and Birmingham City man has scored nine goals in 22 appearances for the eighth-placed Jets.