The president made the request in July during a wide-ranging review of America’s military position after being shown a slide depicting the size of the US nuclear arsenal, three officials have told NBC News.
He is understood to have pointed to the highest number on the chart — about 32,000 nuclear warheads in the late 1960s — and told his advisers that he wanted to increase the current 4,000 warheads to a similar number once more.
Senior advisers told him this would break a wide array of weapons treaties and risk triggering a new global arms race.
Chairman of the joint chiefs of staff Joseph Dunford and defence secretary James Mattis were also at the meeting with Mr Trump and Mr Tillerson.
Mr Tillerson allegedly called the president a moron once the president had left the room.
Mr Trump has dismissed the story as inaccurate, tweeting: “Fake @NBCNews made up a story that I wanted a ‘tenfold’ increase in our US nuclear arsenal. Pure fiction, made up to demean.”
“With all of the Fake News coming out of NBC and the Networks, at what point is it appropriate to challenge their Licence? Bad for country!”
Broadcast licences are administered by the Federal Communications Commission and are not usually revoked unless a holder commits serious illegal conduct.
Mr Mattis also issued a statement last night, describing the report of the alleged request for more nuclear weapons as “absolutely false”.
The “moron” story emerged as the White House grappled with nuclear challenges in North Korea and Iran.
US strategic bombers have carried out exercises close to North Korea hours after Mr Trump, Mr Tillerson, Mr Mattis and a group of senior advisers met to discuss their strategy towards Pyongyang.
The president said: “I think I have a little bit of a different attitude on North Korea than other people might have.
“I listen to everybody but ultimately my attitude is the one that matters isn’t it?”
North Korean foreign minister Ri Yong-ho said Mr Trump’s had “lit the wick of a war against us”.
He told the Russian news agency Tass: “We need to settle the final score, only with a hail of fire, not words.”
Mr Trump is also expected to walk away from the Iran nuclear deal this week, ignoring recommendations from security advisers and allies, including Britain, who fear his decision could spark a Middle East arms race and destabilise the entire region.