White House national security advisor John Bolton said Tuesday that North Korea has not taken the proper steps toward denuclearization, despite an agreement between Pyongyang leader Kim Jong Un and President Donald Trump in June.

Bolton also said Secretary of State Mike Pompeo was prepared to return to North Korea for another meeting with Kim.

"The U.S. has lived up to the Singapore declaration," Bolton said in a Fox News interview Tuesday morning. "It's just North Korea that has not taken the steps that we feel are necessary to denuclearize."

The hawkish advisor vowed that the U.S. would keep up its "maximum pressure" sanctions campaign against Kim's regime until North Korea denuclearizes completely.

"The idea that we're going to relax the sanctions just on North Korea's say-so I think is something that just isn't under consideration," Bolton said.

Bolton's remarks followed recent actions from North Korea, including the return of remains of some Americans killed during the Korean War, that the Trump administration had hailed as signs of progress between the two countries. Vice President Mike Pence flew to Hawaii to receive the 55 sets of remains.

Trump tweeted his appreciation and suggested another meeting with Kim may be forthcoming. "Thank you to Chairman Kim Jong Un for keeping your word & starting the process of sending home the remains of our great and beloved missing fallen!" Trump wrote on Aug. 1. "I look forward to seeing you soon!"

But Bolton took issue with this North Korean gesture, as well. "What a nation that was truly committed to turning the page would do here is return the remains of all the soldiers," he said during the interview.

Trump and Kim traveled to Singapore on June 12 for an unprecedented meeting between a U.S. and North Korean leader. Following the meeting, Trump and Kim signed a joint statement in which the North Korean leader "reaffirmed his firm and unwavering commitment to complete denuclearization of the Korean Peninsula."

Upon his return to the U.S., Trump announced in a Twitter post that "there is no longer a Nuclear Threat from North Korea."

Trump tweet

But The Washington Post reported on July 30, citing officials, that U.S. spy agencies have recently obtained evidence that North Korea is continuing to develop intercontinental ballistic missiles even after the summit.

The National Security Council did not immediately respond to a CNBC request for further comment on Bolton's remarks.

In a separate Fox interview on Monday, Bolton said if Iran wants to avoid the reimposition of U.S. sanctions, it should take up Trump's offer to negotiate.

Asked what the leaders of Iran could do, Bolton said: "They could take up the president's offer to negotiate with them, to give up their ballistic missile and nuclear weapons programs fully and really verifiably not under the onerous terms of the Iran nuclear deal, which really are not satisfactory."

"If Iran were really serious they'd come to the table. We'll find out whether they are or not."

—Reuters contributed to this report.