Seeing a Bloomberg Ad on Fox News, Trump Takes the Bait

But Mr. Bloomberg’s advisers see a void that the candidate is filling, in a primary race during which the candidates have sometimes struggled with whether to depict Mr. Trump as an existential threat or to ignore him altogether. The president, meanwhile, continues to try to set the terms of engagement in the election, threatening to attack them in demeaning, and deeply personal, ways.

Howard Wolfson, an adviser to Mr. Bloomberg, said that Mr. Trump had been “essentially running unopposed until Mike entered the race.”

“While the other Democrats are fighting and sniping with one another in Iowa, we are running a nationwide campaign that is taking the fight directly to President Trump on issues where he is extremely vulnerable,” he said.

As for Mr. Trump’s nickname for Mr. Bloomberg — “Mini” — Mr. Wolfson called it “irrelevant” and said, “If you want to think about small, Trump could fit in Mike’s pocket. The stature gap, the wealth gap, the success gap, the experience gap, the achievement gap between these two men is vast.”

The Bloomberg campaign has been ratcheting up its attacks as Mr. Trump has reacted to the spending.

Guided by extensive internal polling, Mr. Bloomberg’s campaign first began directly attacking Mr. Trump with an ad on health care, accusing the president of ruining insurance for millions of Americans and undermining coverage of pre-existing conditions. The ad about pre-existing conditions immediately drew a Twitter rebuke from the president.

From health care to the environment to impeachment, the Bloomberg campaign has been running ads attacking Mr. Trump’s record nationally, particularly in key swing states. The ad about pre-existing conditions, for example, was backed by more than $1.2 million in the Orlando, Fla., market alone.

But, in a further tweak to Mr. Trump, the campaign is also running ads in his strongholds, such as $14 million worth of ads attacking the president in Texas. The state is also a Super Tuesday state, where Mr. Bloomberg hopes to amass delegates.