US President Donald Trump said Tuesday talks with North Korea were making progress despite reports Kim Jong Uns regime had stepped up its nuclear program after the two leaders held a historic summit.

"Many good conversations with North Korea-it is going well!" he tweeted. "In the meantime, no Rocket Launches or Nuclear Testing in 8 months.

"All of Asia is thrilled. Only the Opposition Party, which includes the Fake News, is complaining. If not for me, we would now be at War with North Korea!"

Trump's comments came as his Secretary of State Mike Pompeo was to head back to North Korea on Thursday to flesh out the bare bones nuclear agreement signed by the two leaders during their summit meeting in Singapore last month.

Trump has boasted that the statement resolved the old foes' nuclear stand-off, but Pompeo has been tasked with nailing down details.

There have been reports Kim's regime has failed to honor its denuclearization pledge after the June 12 meeting.

The Washington Post on Sunday quoted four unnamed US officials who were briefed on intelligence findings that point to preparations for deceiving the United States, while NBC news has said Pyongyang is increasing its nuclear fuel production.

Citing satellite imagery, a respected monitoring group last month said operations and infrastructure works were continuing at the North's main Yongbyon nuclear site, while the Trump administration itself has cited "an unusual and extraordinary threat" from North Korea's nuclear arsenal to extend decade-long sanctions.

A report in Axios on Monday suggested if talks go well, Trump could invite Kim to New York in September to continue the dialogue as the UN General Assembly gets under way.