After sending world No.7 Hubert Hurkacz packing in round two, Thompson took out 30th seed Matteo Arnaldi 7-5 6-2 7-6 (7-5) on day six to storm into the last 16 at Flushing Meadows for a second time.
De Minaur was playing Brit Dan Evans on Saturday night (Sunday AEST), with the winner to face Thompson for a quarter-final berth.
"If he gets through, it would be for sure an Aussie in the quarters. I hope that it is me," Thompson said on court.
"I will be recovering. I won't be looking on. I'll leave that to my coach."
With Alexei Popyrin basking in the glory of his stunning defeat of defending champion Novak Djokovic, Australia has already converted its biggest US Open contingent in 43 years to having at least two players in the men's last 16.
Whether he plays de Minaur or Evans, Thompson will take a power of stopping.
Playing the tennis of his life, having won his maiden ATP title this year in Los Cabos and made the final in Atlanta, the proud Paris Olympian has now matched his career-best grand slam run from the 2020 US Open.
Yet to drop a set this campaign, Thompson fell behind 4-2 in the third against Arnaldi, a 2023 Italian Davis Cup winner widely considered one of the brightest young stars in tennis.
But, as he does, Thompson went straight back to work, breaking Arnaldi in the seventh game and recovering from 3-0 down in the third-set tiebreaker to seal victory after two hours, 33 minutes.
"It was a great match from me. The third round is pressure. I have not been there too much in my career," he said.
"I was getting a little bit agitated there but I dealt with a really well and came out on top. It is really pleasing to win under lights in New York on a Saturday night."
The 30-year-old is set to claim a new career-high ranking inside the world's top 30 and believes he is only now starting to peak.
"It is hard to argue with. I feel like I am getting better as I get older," Thompson said.
"It is definitely the best I have been playing in my career and my results and ranking dictate that."
But Chris O'Connell's Open is over after Australia's fourth third-round representative copped a lesson from world No.1 Jannik Sinner on Arthur Ashe Stadium.
"I felt like he was on from the get-go. I felt a little bit clueless, to be honest," O'Connell said after exiting with a 6-1 6-4 6-2 loss.
"Every single shot, I just felt like I had to do something with it because he was just on me. He was suffocating me.
"I mean, yeah, I've got to believe but I just felt that he was really on fire.
"The best tennis player I've ever played, for sure."
O'Connell can console himself with a career-best payday of $US215,000 ($A$315,000) for reaching the last 32.
Popyrin, the 28th seed, meets world No.20 Frances Tiafoe in the fourth round on Sunday night (9am Monday AEST) in a prime time showdown.
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