A former teammate of Neymar at Brazilian Serie A club Santos, the 27-year-old currently plays for Greek side AEK Athens.
Sportotal.gr reports Rodriguez is "very likely" to join the Jets after positive early discussions between the two clubs.
The diminutive Rodriguez, who stands at 172cm, is an attacking player who can operate centrally or on the left wing.
He has made 25 appearances in the Greek Super League for AEK over the past two seasons and has a strong pedigree in South America.
Rodriguez began his career with top-flight Argentinian side Independiente, playing 87 times before shifting to Santos for the 2012 season.
He played alongside Neymar for two seasons before the Brazilian superstar's transfer to Barcelona.
Rodriguez stayed on but played little first-team football after his first campaign with Santos, going on loan twice before joining AEK last season and laying on two assists on his official debut.
Rodriguez would become Newcastle's third foreign signing and provide new coach Ernie Merrick with another handy weapon in the final third as he seeks to reverse the club's fortunes.
Trying to end a run of seven seasons without making the finals, the Jets open their A-League season on the road against arch-rivals Central Coast Mariners on October 7.