In an interview with the Reuters news agency, he also said he is against a “one-sided” new deal with Russia that would cap the deployment of nuclear warheads.
Mr Trump said: “I am the first one that would like to see everybody – nobody have nukes, but we’re never going to fall behind any country even if it’s a friendly country, we’re never going to fall behind on nuclear power.
“It would be wonderful, a dream would be that no country would have nukes, but if countries are going to have nukes, we’re going to be at the top of the pack.”
The President said the US had fallen behind in its nuclear weapons capacity and also complained about Russian deployment of a cruise missile in violation of an arms control treaty.
He said he would raise the issue with Russia’s President Vladimir Putin when and if they meet.
The new strategic arms limitation treaty, known as New START, between the US and Russia requires that by 5 February 2018, both countries must limit their arsenals of strategic nuclear weapons to equal levels for 10 years.
The treaty permits both countries to have no more than 800 deployed and non-deployed land-based intercontinental and submarine-launched ballistic missile launchers and heavy bombers equipped to carry nuclear weapons, and contains equal limits on other nuclear weapons.
Mr Trump called New START “a one-sided deal”.
“Just another bad deal that the country made, whether it’s START, whether it’s the Iran deal … We’re going to start making good deals,” he said.
In the interview Mr Trump also turned his attention to China, which he said could solve the national security challenge posed by North Korea “very easily if they want to”, ratcheting up pressure on Beijing to exert more influence to stop Pyongyang’s increasingly threatening actions.
He said that while he welcomed China stopping coal imports from North Korea, Beijing needed to put more pressure on Pyongyang.
Speaking from behind his desk in the Oval Office, he also declared himself “very angry” at North Korea’s ballistic missile tests and said accelerating a missile defence system for US allies Japan and South Korea was among many options available.
In a wide-ranging interview, he repeated what he said when he met Israeli Prime Minister Benjamin Netanyahu about liking the concept of a two-state solution for the Palestinian-Israeli conflict – but that he would be satisfied with whatever makes both parties happy.