State Secretary Pompeo made such comments at his briefing to the U.S. Senate Foreign Relations Committee on North Korea on Wednesday.
The impending return of the remains is part of an agreement that President Trump and North Korean leader Kim Jong-un reached during a summit in Singapore in June.
North Korea will transfer the remains of an unspecified number of soldiers killed in the Korean War on Friday, South Korea's Yonhap news agency reported, after accepting about 100 wooden caskets sent by the United States.
The top USA diplomat said Mr Trump "would agree that the primary systems that have threatened America continue to exist".
At the heart of the difficulty, former officials involved in previous recovery missions say, are likely demands from North Korea for cash compensation, as well as the unsolved tensions over North's nuclear weapons and ballistic missile arsenal.
After the historic summit, Trump tweeted that "there is no longer a Nuclear Threat from North Korea", but in recent days he has faced criticism about the lack of clear progress towards denuclearization.
Pompeo said U.S. North Korea policy was guided by a principle stated by Trump on July 17 that "diplomacy and engagement are preferable to conflict and hostility".
U.S. President Donald Trump and North Korea's leader Kim Jong Un leave after signing documents that acknowledge the progress of the talks and pledge to keep momentum going, after their summit at the Capella Hotel on Sentosa island in Singapore June 12, 2018.
A USA military transport plane was set to fly to an airfield in North Korea's northeastern city of Wonsan to accept the remains, Yonhap reported.
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"As you may know, we're also working to bring back the remains of your brothers in arms who gave their lives to Korea, and I hope that very soon these fallen warriors will begin coming home to lay at rest in American soil", Trump told the VFW. He told the lawmakers that they could sleep easy at night.
While South Korean officials say they are committed to the full denuclearisation of North Korea, they have shown more flexibility in the timing of a peace agreement than their US allies.
Pompeo noted that the recovery of remains is unrelated to the main US goal of achieving the "denuclearization" of North Korea. -South Korea alliance with calls for US troops to leave the Korean peninsula, he added. "Otherwise, we could end up paying too high a price for North Korea's trivial denuclearization measures".
He rejected Markey's suggestion that the United States was being "taken for a ride" by North Korea, replying, "fear not senator, fear not".
But Pompeo argued that "progress is happening" with the US's push for denuclearisation.
He said what Trump most likely meant by his statement was that the tension between the US and the North had been "greatly reduced".
South Korea's Unification Minister Cho Myoung-gyon said on Tuesday it is possible to declare an end to war this year.
"For reasons of USA politics, the alliance commitment would effectively end if a peace treaty came about".