Coach Ernie Merrick is a long-time admirer of the J-League-based attacker, the pair developing an alchemy in 2015-2016 when Burns landed the A-League player of the year award under the tutelage of the Scotsman at Wellington Phoenix.
The nomadic 29-year-old is desperate to force his way back into the Socceroos reckoning, a possibility which appears increasingly remote as he remains on the periphery at relegation-threatened Sanfrecce Hiroshima, the club he joined in July from Tokyo FC.
Hence the carrot from the Jets, who are closely monitoring the situation in Japan with a view to possibly bringing Burns back for a second stint in the Hunter during the January transfer window.
After initially seeking to re-sign Burns in the middle of the year, Jets football operations chief and club legend Joel Griffiths said on Sunday: "We're still monitoring his progress just in case there might be a twist or a turn.
"I know how desperate he is to get back into the Australia team.
"But he can't do that without playing. He certainly would get that chance with us.
"We haven't spoken to his agent about him but if something came up regarding his availability we would be mad not to at least have a chat.
"It's strange that he's not been playing in Japan because he clearly still has a lot to offer."
Burns is in danger of be becoming one of Australian football's wasted talents, having played just 32 club games since departing Wellington for FC Tokyo in 2015.
Though he suffered injury setbacks during a five-year span encompassing stints at AEK Athens and K-League Incheon United, just 29 club games during that period tells a story of unfulfilled potential from a player as naturally gifted as any to have emerged since the departure of the 'golden generation'.
Burns, who has 24 caps, last featured for Australia as a 58th minute substitute in a 2-2 draw against Thailand almost a year ago.
His absence since has barely registered a murmur in the light of his continuing lack of club game time.
He is contracted with Hiroshima until mid-2018, and is hoping to feature in the final three games of the season as the club coached by Swede Jan Jonsson fight to avoid the drop.
The unbeaten Jets, meanwhile, are levitating at high altitude after climbing to third on the ladder after five rounds, and might provide a fertile environment for a Burns bounce-back.
"Ernie's a big fan of Nathan's," added Griffiths.
"He helped him to get to Japan with the way he likes to play.
"And obviously helped Nathan helped himself in that regard to.
"Ernie has a lot of praise for him and I think Nathan played the best football all of his career while Wellington.
"He's a good lad too and he would be good for the team.
"He's been here before, obviously under different circumstances.
"We're at club that has changed a lot in terms of personnel and improved infrastructure behind the scenes.
"For us, the right thing to do is not rush into anything (in relation to replacing Vargas)," added Griffiths.
"We get a lot of CVs coming through and it's making sure you tick all the right boxes.
"So we'll take our time and hopefully get somebody in."
The Venezuelan is recovering from surgery on fractured and dislocated ankle suffered against Brisbane Roar in Round 3 and is unlikely to play again this season.