The discussions were expected to go on late into the evening and Hill hinted that they could continue into Friday, although he is due to leave Geneva in the evening.
"I think the plan is to meet early tomorrow morning as well," Hill said.
The US is insisting on receiving a "complete and correct" North Korean disclosure of all its nuclear activities under the deal.
Hill and Kim met in Beijing last month in an effort to break the deadlock, but reportedly achieved no progress.
"I think we should be a little flexible on format, but with the understanding that flexibility on format doesn't mean flexibility on getting a complete and correct declaration," Hill told reporters in Washington before leaving for Switzerland.
He indicated, however, that sensitive elements of any North Korean declaration would not be kept secret.
"I don't think we can have secret agreements, secretly arrived at," Hill said. "We have to make this understandable to the public."
North Korea last year signed a landmark deal to abandon all its nuclear weapons in exchange for badly needed energy and economic aid and major security and diplomatic benefits.
But the process - involving China, Japan, both Koreas, Russia and the US - has been stalled since North Korea missed a deadline at the end of 2007 to declare all its nuclear programmes and disable its plutonium plant.
At stake is a US call for North Korea to clarify its suspected uranium enrichment programme and secret nuclear technology transfers to Syria.
Pyongyang, which conducted its first test of a plutonium-based nuclear bomb in October 2006, has denied the existence of a separate nuclear programme based on a uranium enrichment programme and has rejected alleged links with Syria.
|Hill says the US wants to see a full declaration |
of all North Korea's nuclear activities [Reuters]
Instead North Korea blames Washington for the deadlock, citing a US failure to remove Pyongyang from a list of state sponsors of terrorism.
During a visit to Beijing late last month, Condoleezza Rice, the US secretary of state, urged China - North Korea's main ally - to use "all influence possible with the North Koreans" to end an impasse.
Koh Yu Hwan, a professor at Seoul's Dongkuk University, told the AFP news agency that China might have played a role in resuming the US-North Korea talks.
"China has more leverage than before over North Korea as the North's chronic food shortages become all the more acute this year amid global food price hikes," Koh said.
Earlier this week the US state department reported that North Korea has increased punishment for those trying to flee the desperately poor state.
North Korea controls its population by shutting them off from the world, keeping them in fear through arbitrary and unlawful killings and running a network of political prisons to suppress dissent, the state department said in its annual human rights report released on Tuesday.
The North once sentenced those trying to cross the border in search of food to a few months in a labour camp or simply gave them warnings, "but during the year several sources indicated that the DPRK (North Korea) reversed this more lenient policy," the report said.
Now the reports says would-be refugees face several years in political prison camps if caught trying to flee the country.