"Last weekend Trump claimed that our leadership wouldn't be around much longer and declared a war on our country," Ri said, according to an official translation of his remarks to reporters in New York.
"Since the United States declared war on our country, we will have every right to make all self-defensive counter measures, including the right to shoot down the United States strategic bombers at any time even when they are not yet inside the aerospace border of our country," Ri said.
White House press secretary Sarah Sanders said Monday that the US has not declared war on North Korea, adding, "Frankly, the suggestion of that is absurd."
Sanders said it is "never appropriate" to shoot down another nation's aircraft in international waters and the administration plans to continue to protect the area.
Earlier on Monday, State Department spokesperson Katina Adams told CNN the US seeks a "peaceful denuclearization of the Korean Peninsula."
But the US military "will take all options to make sure that we safeguard our allies and our partners and our homeland so if North Korea does not stop their provocative actions we'll make sure we provide options to the President to deal with North Korea," according to Col. Rob Manning, a Pentagon spokesman.
Asked about Ri's charge that Trump's comments were a declaration of war, Manning, said: "Our job as the Department of Defense is as you know is to make sure that the President is provided military options, we'll continue to do that, and we have a deep arsenal of military options to provide the President so then he can decide how he wants to deal with North Korea and the regime."
"We are postured and we are ready to fight tonight," he added.
The US Navy will also continue to maintain its presence near the Korean peninsula despite the latest round of harsh rhetoric and threats of a military strike from Pyongyang.
Two US defense officials told CNN on Monday that the USS Ronald Reagan aircraft carrier will conduct scheduled naval maritime exercises with the South Korean navy off the peninsula at the end of October.
While the officials said the exercises are long scheduled and not in response to recent tensions, similar exercises have prompted strong condemnation from North Korea in the past.
War of words
The ongoing war of words between the two nations also saw several new fiery salvos on Saturday, a day on which the US military, in a show of force, flew bombers in international airspace over waters east of North Korea and the detection of mysterious seismic activity refueled concerns over the rogue nation's nuclear ambitions.
Speaking at the UN on Saturday, Ri said that Trump had made a missile attack on the US mainland inevitable by insulting the dignity of North Korea.
"None other than Trump himself is on a suicide mission," Ri said in a speech at the UN General Assembly. "In case innocent lives of the US are harmed because of this suicide attack, Trump will be held totally responsible."
Trump took on Twitter Saturday night to respond to Ri's remarks.
"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 wrote.
The US and North Korea exchanged a series of verbal volleys last week as the UN met in New York.
In his speech Tuesday, Trump threatened to "totally destroy" North Korea if the US was forced to defend itself or an ally. He then tweeted on Friday that Kim was "obviously a madman" who would be "tested like never before."
Kim responded to the US President's speech in a rare televised statement in which he said Trump would "pay dearly" for the threats and accused him of being "mentally deranged."
Shortly after Kim's televised address, North Korea's foreign minister said his country could test a powerful nuclear weapon over the Pacific Ocean in response to Trump's threats of military action.
The US military also sent its own message to Pyongyang on Saturday as B-1B Lancer bombers from Guam and F-15C Eagle fighter escorts from Okinawa, Japan, flew in international airspace over waters east of North Korea, according to the Pentagon.
It's the farthest north of the Demilitarized Zone (DMZ) that US fighters or bomber aircraft have ever flown this century, the Pentagon said.
The flight underscored "the seriousness with which we take DPRK's reckless behavior," the Pentagon said, using an acronym for North Korea's official name, the Democratic People's Republic of Korea.
"This mission is a demonstration of US resolve and a clear message that the President has many military options to defeat any threat," Pentagon spokeswoman Dana White said in a statement.
"North Korea's weapons program is a grave threat to the Asia-Pacific region and the entire international community. We are prepared to use the full range of military capabilities to defend the US homeland and our allies."
The bomber flights are a common response to North Korean actions that the United States and its allies perceive as hostile.
A key part of the US military's "tip of the spear," US B-1 bombers have been seen regularly over the Korean Peninsula in recent months amid escalating tensions with Pyongyang -- running regular training flights with Japanese and South Korean fighter jets that often provoke the ire of the North Korean regime.