Although this represents a step forward, as Moscow accepts enlargement will go ahead, some of the conditions raise severe problems for the alliance and pose a big question about the future of Nato's nuclear strategy, an issue ignored for four years.
Nato sources say that Willy Claes, the alliance's secretary-general, was briefed by President Bill Clinton about recent talks between Russia and the US when he visited Washington last week. US and Russian officials met in Washington last month to discuss the proposals.
Russia has put forward several areas of concern over enlargement, which is likely to include Poland, Hungary, the Czech Republic and perhaps Slovakia.
Moscow wants a permanent forum for consultation with Nato that would give it the right to automatic consultation about any security developments in Europe.
Something along these lines is evolving anyway, through bilateral meetings and encounters in the Organisation for Security and Co-operation in Europe (OSCE) based in Vienna.
Russia wants guarantees concerning the way Central European states are brought into Nato. It does not want forward positioning of nuclear weapons in peace time or stationing of alliance forces in these countries. Arrangements such as these exist for Norway and it is understood that there have been discussions on a similar deal for Central Europe. Some prospective new members have indicated they do not want alliance nuclear weapons on their soil.
But this poses big questions for an alliance that is based on the use of nuclear weapons in the last resort.
Most controversially, Russia says it cannot accept the integration of new members into the alliance's military structure. France and Spain are outside the unified command, but by choice. The Central Europeans would strongly oppose such a move, as would Nato. It would leave the Central Europeans with a security guarantee, but no certainty as to how it could be enforced.
Russia also wants changes to an important arms control agreement which the West fears would increase instability in Central Europe. The 1990 treaty on Conventional Forces in Europe (CFE) that limits troops and weapons is seen by the West as an important achievement in arms control. Moscow contends it is outdated and singles out Russia for severe restrictions. Russia wants to replace it with a more general arms-control agreement for Europe that would restrict the number of surprise inspections that can be made on weapons. Moscow also wants limits on military research and development expenditure. And it wants more flexibility over where it can move its troops, Nato and Western diplomatic sources say.
Russia's main concern about the treaty is that it places restrictions on the deployment of Russian forces inside Russia, and in particular in the Caucusus region, including Chechnya.
Western diplomats concede that CFE stigmatises Russia. But they contend that the treaty allows for some flexibility, and they say that there is already a provision for reviewing it in 1996.
Although the West insists that there can be no Russian "conditions" for Nato enlargement, Moscow is clearly putting down markers for the deal that must accompany enlargement. Nato sources say there is a recognition that relations with Central Europe and ties with Russia must move in tandem.