Donald Trump will release “very, very specific” results of a physical examination soon, the Republican presidential candidate said on Monday, a day after the Democratic nominee, Hillary Clinton, was revealed to have pneumonia.

Hillary Clinton cancels California trip after pneumonia forces departure from 9/11 ceremony Read more

“This last week I took a physical and I’ll be releasing when the numbers come in,” Trump said in an interview with Fox News.

Trump has so far released only a short letter from his personal physician, which the doctor in question subsequently said had been rushed. Clinton last year released a longer doctor’s note, written by the same physician who on Sunday released a statement saying that the nominee had been diagnosed with pneumonia on Friday.

“I hope she gets well soon,” Trump said. “I really don’t know what’s going on. Like you, I say what I see. [Clinton’s] coughing fit was a week ago” – at a Labor Day rally in Cleveland – “so that was pneumonia also, I think it would have been, so something’s going on. I hope she gets well and gets back on the trail and we’ll be seeing her at the debate.”

Asked if he thought the Democratic National Committee was preparing a replacement for Clinton, and whether he was ready to contest an election against Clinton running mate Tim Kaine, Trump said: “No, I don’t think they’ll replace her. We have to see what’s wrong, we have to see what’s wrong. But whatever it is, I’m ready.

“I think it’s an issue. In fact, this last week I took a physical and I’ll be releasing when the numbers come in. Hopefully they’re going to be good, I think they’re going to be good, I feel great, but when the numbers come in I’ll be releasing very, very specific numbers.

“I’ve already done it but the report should be finished this week.”

Trump chose to focus his political fire on Clinton’s remark at a Friday fundraiser in New York that “to just be grossly generalistic, you could put half of Trump’s supporters into what I call the basket of deplorables”.

Trump said he had initially thought “it was not within the realm of possibility that she said it” and called the remark “the biggest mistake of the political season”. “It was said with such anger,” he added.

Clinton issued a statement over the remark on Saturday, in which she repeated that she had been being “grossly generalistic” but said she had been wrong to say a “half”.

On Sunday, Clinton left 9/11 commemorations in New York after an hour and a half, and was later seen in video shot by a bystander stumbling as she was supported into a car. Amid press confusion and reports of a “medical episode”, she went to her daughter’s Manhattan apartment to rest. Her campaign initially said she had been “feeling overheated”; Clinton emerged from the apartment to say she was “feeling great”.

“Secretary Clinton has been experiencing a cough related to allergies,” Dr Lisa R Bardack said in a subsequent statement, having examined Clinton at her home in Chappaqua. “On Friday, during follow-up evaluation of her prolonged cough, she was diagnosed with pneumonia. She was put on antibiotics, and advised to rest and modify her schedule.

“While at this morning’s event, she became overheated and dehydrated. I have just examined her and she is now rehydrated and recovering nicely.”

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Clinton was ordered to rest and subsequently canceled a Monday trip to a fundraiser in California, although the campaign said she planned to appear by video instead.

Trump is due to appear on the Dr Oz television show this week, to discuss the health of both presidential candidates. The candidate himself and his campaign surrogates have cast doubt on Clinton’s health and fitness for the Oval Office, based on a 2012 incident in which Clinton fell, a mishap attributed to a stomach virus, and suffered a concussion and a blood clot in the brain. Later testing showed the clot to have cleared completely.

The release of health records – like the release of tax returns, which Trump has not done – is traditional rather than mandatory. In 2008, Barack Obama, then 47, released a 276-word report about his health. His opponent, John McCain, then 71, made available more than 1,000 pages related to his own medical history.

A spokesman for Gary Johnson, the Libertarian nominee, said in a statement to the Guardian on Sunday: “Given Governor Johnson’s level of fitness and exercise, his medical records haven’t been much of an issue. We will discuss with him how and what information to release.”