Tensions between US President Donald Trump and Zohran Mamdani showed no signs of easing on Wednesday as both hit out at each other's political approach, just hours after the 34-year-old democratic socialist was elected as New York City's next mayor.

"Donald Trump, since I know you're watching, I have four words for you: Turn the volume up," Mamdani, a Democrat, told the Republican president from the stage of his Brooklyn victory party.

Trump was watching, White House Press Secretary Karoline Leavitt confirmed.

Trump was born in New York in 1946, and Mamdani said: "If anyone can show a nation betrayed by Donald Trump how to defeat him, it is the city that gave rise to him."

Trump says US has lost 'sovereignty' after Mamdani win

On Wednesday, Trump repeatedly referred to Mamdani while speaking at a business conference in Miami.

Trump said the US had lost "sovereignty" in the wake of Mamdani's election victory.

The president said Miami "will soon be the refuge for those fleeing communism in New York."

"The decision facing all Americans could not be more clear: We have a choice between communism and common sense," he said, while adding that the Democrats offered an "economic nightmare" and his policies would provide an "economic miracle."

The speech marked the first anniversary of Trump's presidential election win over Democrat Kamala Harris.

"We rescued our economy, regained our liberty, and together we saved our country on that magnificent night 365 days ago," Trump told his audience of supporters.

But Trump was not finished with the pointed remarks aimed at New York's mayor-elect, suggesting Mamdani was indicative of a wave of Democrats in Washington.

"If you want to see what congressional Democrats wish to do to America, just look at the result of yesterday's election in New York where their party installed a communist as the mayor of the largest city in the nation," Trump said.

Trump threatens to pull the plug on New York City funding

Separately, in an interview with US broadcaster Fox News, Trump said he is willing to talk with Mamdani, but emphasized that the new mayor needs to be respectful of Washington — and its financial support of New York City — if he wants to succeed.

Trump spent months trying to undermine the Mamdani campaign. 

He also threw his weight behind the mayoral bid of former Governor Andrew Cuomo, a former Democrat himself who ran as an independent after losing to Mamdani in the Democratic primary in June.

In supporting Cuomo, Trump bypassed the Republican candidate Curtis Sliwa. Trump said a vote for Sliwa would only help Mamdani get elected, urging New Yorkers to vote tactically to keep Mamdani out of office.

Why does Mamdani attract Trump's ire?

Mamdani, who was born in Uganda and became a naturalized American citizen after graduating from college, cast himself as the embodiment of the resistance against Trump, who has pursued an anti-immigrant agenda since being sworn in as president in January.

"New York will remain a city of immigrants, a city built by immigrants, powered by immigrants and, as of tonight, led by an immigrant," Mamdani said in his election night speech. "So hear me, President Trump, when I say this: To get to any of us, you will have to get through all of us."

Edited by: Srinivas Mazumdaru