'We are encouraged by the way things are going. We think we are making progress,' said the US spokesman, Richard Boucher.
The ministers held a two-hour morning session and had lunch together before handing the negotiations over to experts. They agreed to continue talks over dinner and to meet again today, Mr Boucher said.
When the talks began, Mr Kozyrev sounded optimistic about the prospects of sealing the treaty, known as Start-2. 'I am ready to bet a bottle of whisky and say that we will do such work here which will make it possible for our Presidents to have a positive decision after our negotiations today,' he said.
The ministers are making a final effort to complete the treaty before the Bush administration leaves office on 20 January. It would slash both countries' arsenals of long-range nuclear weapons by about two-thirds.
Flying to Geneva from Washington, Mr Eagleburger said he thought there was a 'better than 50-50 chance' of completing the treaty which would give Mr Bush a triumphant exit from office.
The first Strategic Arms Reduction Treaty (Start), fixed last year, cut the nuclear arsenals of the US and former Soviet Union by about 30 per cent. Start-2 goes much further. It would completely abolish the most deadly and destabilising class of nuclear weapons - land-based missiles equipped with multiple warheads.
If the talks succeed, Mr Bush and Russia's President, Boris Yeltsin, are expected to meet next week in the Black Sea resort of Sochi to sign the treaty.
Mr Eagleburger said the United States had made some important concessions in seeking to conclude the deal and Moscow was also showing flexibility. Three major issues remain to be resolved:
How many silos that currently house the 154 giant SS-18 missiles would the Russians have to destroy and how many would they be allowed to keep?
How many of its 170 mobile SS-19 missiles would Russia be allowed to convert or 'download' from six warheads to one so as to be legal under the treaty?
The fate of the United States' B-1 and B-52 bombers.