North Korea's foreign minister says President Donald Trump had declared war on his country and that Pyongyang reserves the right to take countermeasures, including shooting down US bombers even if they are not in its airspace.

The increasingly heated rhetoric between Trump and North Korean leader Kim Jong-un is raising fears of a risk of a miscalculation by one side or the other that could have massive repercussions.

"The whole world should clearly remember it was the US who first declared war on our country," Ri Yong-ho said in New York City on Monday.

"Since the United States declared war on our country, we will have every right to make countermeasures, including the right to shoot down United States strategic bombers even when they are not inside the airspace border of our country."

In a direct reference to a Twitter post by Trump on Saturday, Ri said: "The question of who won't be around much longer will be answered then."

US stocks fell sharply in late morning trading on Monday after Ri's comments. The five tech companies - Facebook, Amazon, Apple, Netflix and Alphabet - were down between 3.7 percent and 1.05 percent.

'Mr Evil President'

Ri told the UN General Assembly on Saturday that targeting the US mainland with its rockets was inevitable after "Mr Evil President" Trump called Kim a "rocket man" on a suicide mission.

"Just heard Foreign Minister of North Korea speak at UN. If he echoes thoughts of Little Rocket Man, they won't be around much longer!" Trump said on Twitter on Saturday.

WATCH: Inside the war of words - Covering the North Korean threat

Al Jazeera's Diplomatic Editor James Bays, reporting from New York, said the latest comments by Ri represent a "real ramping up" of language against the US.

"He said if US bombers were to fly near North Korea even in international waters, North Korea will shoot them down. The minister's comments on Saturday at the UN General Assembly were a blistering attack, but he's gone even further now."

North Korea, which has pursued its missile and nuclear programmes in defiance of international condemnation and economic sanctions, said it "bitterly condemned the reckless remarks" of Trump.

They were an "intolerable insult to the Korean people", and a declaration of war, North Korea's official news agency said on Monday.

Pyongyang accuses Washington, which has 28,500 troops in South Korea, a legacy of the 1950-53 Korean War, of planning to invade and regularly threatens to destroy it and its Asian allies.

The US and South Korea are technically still at war with North Korea because the 1950s conflict ended with a truce, not a peace treaty.

Kim's broadside

In a rare direct statement on Friday, Kim described Trump as a "mentally deranged US dotard" whom he would tame with fire.

Kim said North Korea would consider the "highest level of hardline countermeasure in history" against the US and that Trump's comments had confirmed his nuclear programme was "the correct path".

Trump threatened in his maiden UN address last Tuesday to "totally destroy" North Korea if it threatened the US or its allies.

For its part, China called on Monday for all sides in the North Korea missile crisis to show restraint and not "add oil to the flames".

Asked how concerned China was the war of words between Trump and North Korea could get out of control, Lu Kang, China's foreign ministry spokesman, described the situation as highly complex and sensitive.

It was vitally important that everyone strictly, fully and correctly implemented all North Korea related UN resolutions, Lu said, resolutions which call for both tighter sanctions and efforts to resume dialogue.

All sides should "not further irritate each other and add oil to the flames of the tense situation on the peninsula at present", Lu told a daily news briefing.

"We hope all sides do not continue doing things to irritate each other and should instead exercise restraint."

Hydrogen bomb threat

North Korea conducted its sixth and most powerful nuclear bomb test on September 3. Pyongyang said on Friday it might test a hydrogen bomb over the Pacific Ocean.

While China has been angered by North Korea's repeated nuclear and missile tests, it has also called for the US and its allies to help lessen tension by scaling back their military drills.

US Air Force B-1B Lancer bombers escorted by fighters flew in international airspace over waters east of North Korea on Saturday in a show of force the Pentagon said indicated the range of military options available to Trump.

In response to a question about the exercises, Chinese spokesman Lu said: "A continued rise in tensions on the peninsula, I believe, is not in the interests of any side."

Wang Jingdong, president of the world's largest lender Industrial and Commercial Bank of China (ICBC), said during a briefing the bank would "strictly implement UN Security Council decisions related to North Korea and carefully fulfil relevant international responsibility".