The communication comes as North Korea edges towards a deadline to "disable" its key nuclear complex at Yongbyon and disclose all its nuclear activities by the end of December.
Under the February 13 agreement reached at talks with the US, South Korea, Japan, Russia and China, North Korea agreed to end its nuclear programme in exchange for heavy fuel oil or equivalent aid.
Hajime Izumi, a Korea expert at the University of Shizuoka near Tokyo, said he believed Bush's letter stressed the US position that it would forge diplomatic relations with North Korea if Pyongyang completely dismantled its nuclear programmes.
"Anything else is unthinkable," he said.
Hill said the North was moving to cripple the reactor and other units at Yongbyon so they would be difficult to restart.
But disagreements remain over what should appear in the comprehensive declaration of nuclear activities that Pyongyang has promised.
Before meeting Chinese diplomats to brief them on his Pyongyang trip, Hill said that one of the points of dispute was North Korea's efforts to enrich uranium, a way of making nuclear material that does not rely on reactors.
Hill said: "We've had a lot of discussions with them about uranium enrichment."
He said that the United States had "very good evidence" that North Korea had bought enrichment technology and had received assistance from Pakistan.
The February agreement would still leave North Korea to take the crucial steps of irreversibly dismantling Yongbyon and handing over any nuclear weapons materials.
But Pyongyang may now miss the year-end deadline for the disarmament steps, South Korea's foreign minister indicated.
According to a spokesman for his ministry, Song Min-soon said: "We are aiming for the initial end-of-the-year deadline, but we may need to be a little more flexible."
Recently, the Bush administration has played down claims about how advanced North Korea was in enrichment.
However, Hill suggested that even if those efforts were fruitless or dormant, North Korea had to make a full disclosure.
He said: "We want to be completely sure they don't have any ongoing programme. Being clear about what's happened is also a means for us to build a future relationship."