Michael Bloomberg traded insults with Donald Trump on Sunday, calling the president “a pathological liar who lies about everything: his fake hair, his obesity and his spray-on tan”.

The comment, made through the Democratic presidential candidate’s press secretary, came in response to abusive tweets and remarks the president made in a Super Bowl interview with the Fox News host Sean Hannity.

In a quickfire section of the interview, Trump was asked what he thought of when he heard the names of various political rivals.

“Very little,” he said of Bloomberg. “I just think of little.”

Bloomberg, 77, is 5ft 8in tall. Trump, 73, is 6ft 3in.

Trump then repeated an evidence-free claim, also made on Twitter earlier, about Bloomberg demanding special treatment in the Democratic presidential primary because of his height.

“Now he wants a box for the debates to stand on. OK, it’s OK, there’s nothing wrong, you can be short. Why he should he get a box to stand on, OK? He wants a box for the debates. Why should he be entitled to that, really? Does that mean everyone else gets a box?”

The president also said he would “love to run against Bloomberg. I would love it.”

Bloomberg told reporters in Los Angeles he would “stand twice as tall as he does on the stage that matters” and said Trump “knows that I can beat him” .

An undignified exchange with Trump can be a sign of a candidate gaining credibility, at least with a president notoriously prone to lash out under pressure.

Since declaring his run for the Democratic nomination in late November, Bloomberg has spent liberally on advertising and campaign infrastructure. He has begun to make a corresponding impact on the race and the polls.

He is not competing in Iowa, which votes on Monday, or New Hampshire which follows. But he is now fourth in the realclearpolitics.com national polling average and this week the Democratic National Committee (DNC) changed debate qualification rules which had kept his self-financed campaign out of the TV spotlight.

As a result, Bloomberg looks set to appear in the first debate under the new rules, in Nevada on 19 February.

Trump broadened his attack on Twitter.

“Mini Mike is part of the Fake News,” the president wrote early on Sunday, using his nickname for the owner of Bloomberg News.

Trump regularly complains about mainstream coverage of his administration. Bloomberg has not divested from his businesses and questions about how his media company will cover his campaign have swirled from the off.

“They are all working together,” Trump said. “In fact, Bloomberg isn’t covering himself (too boring to do), or other Dems. Only Trump. That sounds fair! It’s all the Fake News Media, and that’s why nobody believes in them any more.”

After Bloomberg declared his White House run, the editor-in-chief of Bloomberg News said the company would not investigate Bloomberg or any other Democrat but would still investigate Trump. The decision met with widespread criticism.

Trump also said: “Many of the ads you are watching were paid for by Mini Mike Bloomberg. He is going nowhere, just wasting his money.”

The former mayor clapped back, tweeting: “Looks like our ads are keeping you up at night. We’ve got one in particular you should watch today.”

That was a reference to an ad that will run during the Super Bowl, focusing on Bloomberg’s push for gun safety reform. Trump will also run an ad during the game, which attracts a massive TV audience.

Trump also claimed Bloomberg was “getting the DNC to rig the election against Crazy Bernie [Sanders], something they wouldn’t do for … others. They are doing it to Bernie again, 2016.”

Sanders challenged Hillary Clinton unexpectedly strongly four years ago, in many eyes damaging her before her defeat by Trump.

Many Sanders supporters claimed the primary was rigged. The issue raised its head again this week when Politico reported that “a small group of DNC members has privately begun gauging support for a plan to potentially weaken Bernie Sanders’ presidential campaign”.

Many party figures question the desirability of a Sanders nomination, in part because the Vermont senator is not a registered Democrat, a point which can rebound on Bloomberg given he switched from Democrat to Republican to run for mayor in New York.

Bloomberg’s national press secretary, Julie Wood, issued the insulting response to Trump’s tweets. The candidate himself told reporters: “The president lies, he lies about everything, so it shouldn’t be a surprise that he said things like that.

“I will stand on my accomplishments of what I’ve done to bring this country together and get things done. I’ve been doing it for a long time and I stand twice as tall as he does on the stage that matters.

“This is what happens when someone like me rises in the polls. All of a sudden the other candidates get scared and I think Donald Trump knows that I can beat him.”

Bloomberg has previously needled the president about their respective fortunes. According to Forbes, Bloomberg is worth around $60bn, 17 times as much as Trump.