The White House is seeking an extra $4 billion for a missile defence in the face of growing threats from North Korea.

The money would be used for a system to bolster the ability to "detect, defeat, and defend against any North Korean use of ballistic missiles against the United States, its deployed forces, allies or partners," President Donald Trump said in a letter to Congress seeking the money.

The supplemental budget request, which was sent to Congress on Monday, also seeks $1.2 billion more for the Trump administration's new Afghanistan strategy - and nearly $700 million to repair two Navy ships badly damaged in separate fatal collisions this year.

The letter also reiterates the president's desire for Congress to pass $1.6 billion in funding to build a wall on the border with Mexico.

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"Providing for the safety and security of the American people is my top priority," Trump wrote in the letter.

"That priority is reflected in both the enclosed [Department of Defense] Budget amendments and the border wall request, which provides the down payment on what [Customs and Border Protection] needs to secure the southwest border."

The missile defence spending would also pay for 16 Standard Missile-3 Block IIA interceptors, 50 Terminal High Altitude Area Defense interceptors, missile detection radar upgrades, intelligence and reconnaissance capabilities, and long-range strike capabilities, among other areas.

Meanwhile, Trump struck a hard line against North Korea's nuclear weapons program and urged Japan to do the same as he closed out two days of talks, dinner and golf diplomacy with ally Japan.

The president refused to rule out eventual military action against the north and exhorted dictator Kim Jong Un to stop weapons testing, calling the recent launches of missiles over Japanese territory "a threat to the civilized world and international peace and stability."

"We will not stand for that," Trump said at news conference with Japanese Prime Minister Shinzo Abe.

"The era of strategic patience is over. Some people say my rhetoric is very strong but look what has happened with very weak rhetoric in the last 25 years."