Voters overwhelmingly reject the idea of a Trump dynasty, with 61 per cent saying neither Mr Trump Jr, 42, nor Ms Trump, 38, should not run.
In findings that are likely to disappoint Mr Trump Jr, in particular, only 16 per cent of American voters thought he should dedicate himself to politics.
Among Trump voters the support was only slightly better — only 35 per cent thought he should run, while 31 per cent disagreed.
Ms Trump’s prospects, among Trump supporters, were even worse: only 31 per cent felt she should continue in her father’s footsteps, while 36 per cent thought it was a bad idea.
The Independent commissioned JL Partners for the survey, and they spoke to 1,002 Americans from 26 October to 28 October.
Speculation has been growing about Mr Trump Jr’s ambitions.
He has been one of the most active campaigners for his father’s re-election and has thrown himself into politics, relishing the bloody battles and frequently taking to Fox News to savage Mr Trump’s critics.
While his sister has been working inside the White House, and has been telling friends she is looking forward to returning to her previous life in New York, Mr Trump Jr has loved lobbing social media grenades from the sidelines, unencumbered by official political constraints.
Last week when he saw a campaign sign in Nevada reading: “Don Jr 2024” he was so delighted he posted it on Instagram, saying: “This will make the lib heads explode. To whomever made that thanks for the compliment... but let’s get through 2020 with a big win first!!!!”
Some have suggested he may dip his toes into an official run in a rural red state like Idaho or Montana, where his love of hunting and his surname will both stand him in good stead.
“If we see Trump getting voted out on 3 November, I don’t think it’ll be the end of Trumpism,” said Rob Goldstone, a PR who worked with the family, famously setting up a 2016 meeting in Trump Tower with Mr Trump Jr and a Russian lawyer.
“I think we’ll see Don Jr go into politics - he loves to hunt, and he’s had a taste of blood,” he told the Independent.
"If anyone was to enter politics it’d be him.
“The other one who may is Lara Trump, Eric’s wife, at a local level. Trump has said she carried North Carolina — where she’s from — and she’s a broadcaster who worked on TV before her marriage, so has been pretty effective. I don’t see her walking away.”
The Independent poll also revealed how the First Family’s popularity has plummeted in last four years.
Thirty eight per cent of people who responded said they felt worse about Jared Kushner than they did in 2016, and only 14 per cent said he had gone up in their estimation.
His wife fared even worse: 43 per cent said their view of her had worsened, and 23 per cent said they felt better.
For Don Jr, 30 per cent feel better about him but 50 per cent say they feel worse.