Talks on Russian entry into the World Trade Organization have been making slow progress on the eve of the annual G8 summit in St Petersburg.
Bush and Putin say personal ties are warm
The issue is high on the agenda of US President George W Bush and his host, Vladimir Putin, who met on Saturday as other leaders were due to arrive.
Mr Bush said a trade deal had been "almost reached" and talks will go on.
Other key issues for the summit are energy security and the crisis in the Middle East.
Mr Bush arrived in Russia's second city on Friday - the full G8 summit will officially open on Sunday morning.
Speaking after the talks, the two leaders said they had reached agreement on global nuclear energy, and fighting nuclear and other forms of terrorism.
On the violence in the Middle East, Mr Bush called on Hezbollah to lay down its arms while Mr Putin called for a "balanced" use of force in the Israeli-Lebanese crisis.
No deal has yet been struck on Russia joining the WTO, US and Russian officials confirmed, though a US spokesman reported "significant progress".
G8 SUMMIT: 15-17 JULY
The world's seven richest nations - the US, Japan, Germany, France, Britain, Italy and Canada have met annually since 1975
Russia joined in 1998, turning the G7 into the G8
2006 summit to be held in St Petersburg - the first time Russia has hosted the G8
Energy security, infectious diseases and education are on Russia's agenda
Mid-East crisis, Iran, North Korea, and international terrorism are also likely to be discussed
Russia is by far the largest economy still outside the 149-member group but the US still has some concerns - particularly on intellectual property rights.
Despite a thaw in the run-up to the summit, relations between the US and Russia have been frosty over recent months.
Mr Bush has pledged to raise concerns about freedom in Russia during their meetings but said he would not lecture Mr Putin.
President Putin recently accused Western critics of Russian democracy of using "colonialist" rhetoric and said they should not interfere in Russia's affairs.
The two leaders dined on Friday night at the opulent 18th-Century Constantine Palace, which will host the G8 meetings.
On Saturday, Mr Bush thanked his host for a "good dinner" and the 60th birthday present he received.
"Our relationship is good," he added.
Seven of the G8 nations, uneasy about the increasing role that Russia will have in meeting their energy needs, want energy supplies to be secure, says BBC economics correspondent Andrew Walker.
Russia in turn wants its customers to be secure.
The debate will take place in the context of new highs in oil prices, which have been given a further upward push by the latest fighting in the Middle East.
The BBC's Jonathan Marcus in St Petersburg says a resurgent Russia is eager to play a role on the world stage and the G8 summit provides just such a platform for Mr Putin.
But our correspondent says that for the Americans there is a fundamental ambivalence in their dealings with Moscow.
Mr Bush needs to keep his relations with Mr Putin on an even level but many in the US still question what they see as failures in Russian democracy.