The US president, speaking in London at the start of the Nato summit, claimed he would "stay out" of the General Election campaign because he does not want to "complicate it".
But Mr Trump went on to describe Boris Johnson as "very capable" and said he would be meeting the Prime Minister during the course of his visit to the UK.
In a breakfast meeting with Nato Secretary General Jens Stoltenberg at the US ambassador's residence in London, Mr Trump was asked if the NHS should be on the table in trade talks.
He replied: "No, not at all, I have nothing to do with it. Never even thought about it, honestly."
The president added: "I don't even know where that rumour started. We have absolutely nothing to do with it and we wouldn't want to if you handed it to us on a silver platter, we want nothing to do with it."
Labour leader Jeremy Corbyn published a letter to Mr Trump on Monday night in which he asked for "reassurances" that US negotiators would not look to push up UK medicine prices by seeking access to the NHS for major American pharmaceutical companies.
His party has repeatedly claimed that the health service is "on the table" in a post-Brexit trade deal with the US.
Mr Trump said he could "work with anybody" when asked whether he could work with Mr Corbyn as prime minister, though earlier said he thought Mr Johnson would do a "good job".
Mr Trump was asked about the Duke of York stepping down from royal duties, and he said: "I don't know Prince Andrew, but that's a tough story, it's a very tough story. I don't know."
Leaders including Mr Trump and Mr Corbyn will gather for a reception at Buckingham Palace tonight.
Mr Corbyn said today that he would tell the President he does not want US companies running the NHS if they meet at Buckingham Palace during the Nato reception.
Asked what he would say under the circumstances, during an interview after launching his workers manifesto, the Labour leader said: "That obviously we want a good relationship with the USA but the trade talks that have been undertaken by the Government have been done in secret and we do not accept any idea of US companies coming in to run our NHS or the idea of extending the patents of medicine that will increase the costs for our NHS."
Asked what he would say to Mr Trump over security matters and Nato, Mr Corbyn said: "That we need to live in a world of peace, we need to live in a world that includes dialogue and we need to improve the dialogue with Russia whilst at the same time being critical of their human rights record."
Last week, the Labour Party leader called a press conference at which he brandished an unredacted report that gave details of meetings between US and UK officials, where they discussed the stipulations of a free trade deal between the two nations after Britain leaves the European Union.
The document included confirmation of a round of meetings "dedicated solely to patents and pharmaceuticals", where officials explained how drugs were approved for use on the NHS and described a US request for "total market access" to UK public services - a form of privatisation - as a "baseline" for an agreement.
Mr Corbyn today twice declined to defend the provenance of Labour's leaked documents on US-UK trade talks after parallels were drawn with the disclosure and a Russian misinformation campaign.
Asked where they came from after it was suggested they may have made it to Reddit via Russian hackers, the Labour leader said: "This is such nonsense, such utter nonsense. This is desperation by the Conservative Party.
"These documents represent secret talks between the Tory Government and the American administration to sell off our public services to American corporations. Not even the Government has claimed they are false documents, they have challenged our analysis of them, they've challenged what we think they are trying to do on them but they haven't challenged the honesty of them.
"So I think this is just such nonsense."
Pressed if he is sure of the source from which Labour got the documents, he said: "Those documents represent meetings that took place between a British minister and officials in the USA and a lot more meetings that took place between British officials and US officials to discuss how they would open up our public services to American corporations.
"I think it is time they give us an assurance that they will not sell off our NHS, that they will not sell off our public services as part of a trade deal with the USA."