The two leaders held their first summit since Hu became president earlier this year at the lakeside setting of Evian, on the sidelines of the Group of Eight summit of industrialised nations.
Hours earlier Bush met Russian President Vladimir Putin in Saint Petersburg, as part of a series of US consultations with Asian powers, including Japan and South Korea on what to do about Pyongyang's nuclear drive.
Hu passed a message to Bush from North Korea, calling for him to agree to "bilateral contacts" with Pyongyang in return for it joining a multi-nation dialogue on the showdown, which erupted last year.
But a senior US official told reporters that Bush stuck to his position that muliti-lateral talks were the only way forward, amid signs of horsetrading on the format for future talks, after three-way talks in Beijing in April.
"The Chinese did tell us that the North Koreans are pushing for a bilateral conversation," said the official, on condition of anonymity.
Bush, who has adopted a hard-line policy towards North Korea, ruled out separate contacts with the Stalinist state, but indicated he may accept a statement directed to the US in the presence of other states.
"Within those multilateral talks, if the North Koreans look us in the eye on one corner of the table and say things directly to us, we're going to listen, obviously and will do the same thing back to them".
"We're not going to go into a separate room. We're not going to say, please North Korea, you come and talk - spill to us without the Chinese listening to what's going on here".
The US has imposed intense pressure on China over North Korea, as it is seen as one of the few states with any influence over Pyongyang.
Bush invited Hu to visit Washington, and praised his opposite number's leadership during the SARS crisis, the senior US official said.
"President Bush issued an invitation to President Hu to visit Washington as soon as he can make it, hopefully perhaps in late 2003 or early 2004," said the official on condition of anonymity.
Bush also praised Hu for coming clean on China's SARS crisis after earlier complaints that Beijing had tried to cover up the extent of the epidemic.
"President Bush once again expressed condolences to the families and friends of SARS victims in China and specifically praised President Hu's leadership in addressing this problem," said the official.
"He specifically praised President Hu's willingness to become transparent on the issue of SARS".
Latest figures put the number of people infected with SARS on the Chinese mainland at 5 328 with 332 deaths.
The senior official said Bush also welcomed China's willingness to sign within the next few weeks an agreement on the US container security initiative, designed to safeguard global freight flows from terrorists.
He also stressed the importance of US-China cooperation on proliferation of components for weapons of mass destruction, and requested that China make more of an effort to enforce regulations in the area.
The US last month slapped sanctions on two major arms companies from China and Iran, accusing them of working in concert to help the Islamic government in Tehran modernise and expand its missile arsenal. – Sapa-AFP.