In a letter to House Speaker Paul Ryan, Trump also requested an additional $700 million to repair damage to US Navy ships and $1.2 billion in support of the administration's South Asia strategy.
"This request supports additional efforts to detect, defeat, and defend against any North Korean use of ballistic missiles against the United States, its deployed forces, allies, or partners," the letter said.
An additional $700 million would be used to "repair damage to and restore the operational readiness of the USS John S. McCain and the USS Fitzgerald," according to the letter.
"These ships provide critical naval presence and additional ballistic missile defense capabilities in the Asia-Pacific theater," the letter said.
The guided-missile destroyers USS Fitzgerald and USS John S. McCain
collided with commercial ships in June and August, respectively, and resulted in the deaths of 17 sailors.
The $1.2 billion requested in support of the Trump administration's South Asia strategy would largely go toward funding increased troop levels in Afghanistan -- allowing for the deployment of an additional 3,500 US troops to the region and special operations forces capabilities.
Earlier this year, the Trump administration requested a $54 billion defense budget increase aimed at adding more ships and aircraft.
Trump's request comes during his trip to Asia, where escalating tensions with North Korea have been a focal point in meetings with various allies in the region, including South Korea and Japan.
Despite asking for additional funding to beef up American ballistic missile defense capabilities in the region, Trump said Tuesday that he believes US-led international efforts to crank up the pressure on North Korea and his at-times menacing rhetoric are beginning to work.
The US President did not offer any specific signs of progress, noting that his administration likes "to play our cards a little bit close to the vest." But alongside South Korean President Moon Jae-in during a joint news conference, Trump said he believes the US-led coalition is "making a lot of progress."
The comments marked a rare time in which Trump has signaled US efforts to counter the North Korean threat are yielding positive results. And while he did not signal the military option had left the table, Trump on Tuesday leaned more heavily into hopes for a diplomatic solution rather than the issuance of threats of military action.
Trump left no doubt that the United States would use military force if necessary to protect itself and allies in Asia from North Korean threats.
"The United States stands prepared to defend itself and its allies using the full range of our unmatched military capabilities if need be," Trump said.
Trump advocated a more militaristic Japan on Monday, saying Tokyo should begin purchasing American-made military equipment to better protect itself against threats from North Korea.
Responding to reports he wanted Japan to shoot down a North Korean missile that passed over its territory, Trump said the country should be protecting itself.
"He will shoot them out of the sky when he completes the purchase of a lot of military equipment from the United States," Trump said of Japan Prime Minister Shinzo Abe, with whom he sat for talks earlier in the day.
Abe responded by saying his government already buys a lot of military equipment from the US, but agreed with Trump the country needs "to enhance our defense capability."