The comments came after Pyongyang on Sunday triggered global alarm with its most powerful nuclear blast to date, claiming to have successfully tested a hydrogen bomb.
Putin condemned the test and said it was a "provocative" act.
"The United States will look at every country that does business with North Korea as a country that is giving aid to their reckless and risky nuclear intentions", she said.
It would also impose a complete ban on the hiring and payment of North Korean laborers overseas, estimated to be between 60,000 and 100,000.
In addition the United States said it may stop trading with any country that does business with North Korea.
"The worldwide community must unite in applying the greatest possible pressure on North Korea", he said just four days after Pyongyang staged its sixth and most powerful nuclear test to date, which it described as a "perfect success".
But Moon, the South Korean president, has asked the United Nations to consider blocking oil shipments to the North, government officials in Seoul told reporters. By comparison, China provides it with about 520,000 tonnes of crude a year, according to industry sources.
Beijing military expert Li Jie told South China Morning Post that the drill shows that China strongly condemns the nuclear test.
Also on Sunday, Trump mentioned the possibility of "stopping all trade with any country doing business with North Korea".
Developing such a weapon would significantly bolster South Korea's capacity for action independent of its ally the U.S., which has kept a strong military presence in the country since the Korean War and has its own arsenal of "bunker buster" bombs with payloads of up to 2,400kg (5,300lbs).
Speaking after the talks, the two men said that resolving the North Korea crisis as soon as possible was their top priority but they appeared to have diverging views on the problem.
Insisting on the futility of the new sanctions, Pyongyang said it will respond "to the heinous sanctions and pressure of the USA with our own mode of counter offensive" and Washington "shall be held totally responsible for all catastrophic consequences to follow".
The council last month imposed new sanctions on the country over two long-range missile launches in July.
Ministry spokesman Geng Shuang clarified later on Thursday that China would support the consensus of the UN Security Council.
Asian stocks fell on Wednesday after a slide on Wall Street overnight while the dollar was on the defensive with Korean tension showing few signs of abating.
Fears of a leak from North Korea's detonation of what it claimed was a hydrogen bomb came after Chinese monitors detected a second tremor shortly after the initial quake triggered by the blast.