The two sides have already resolved all of Russia's concerns over the economic impact of enlargement. Last week, they reached agreement on most key issues to do with goods transit between the Russian mainland and Kaliningrad.
Last night, the EU agreed to what it hopes is an acceptable compromise for Russia to address what is perhaps the thorniest issue -- the situation of the Russian-speaking minorities in Estonia and Latvia.
There are hundreds of thousands of Russian speakers in the two Baltic countries. Moscow has argued that the Russian-speaking minority, though large, suffers discrimination because of strict language and citizenship laws that affect everything from education to employment.
Russian Foreign Minister Sergei Lavrov is due today to hold talks on the issue with European Union officials. If Russia accepts the compromise, it will pave the way for extending Moscow's partnership treaty with the EU to the bloc's new members.
This in turn will help prevent a possible trade war and heightened tensions in EU-Russia relations.
According to a draft obtained by RFE/RL, the EU now suggests that the last paragraph of its joint statement with Russia should read as follows: "The EU and the Russian Federation welcome EU membership as a firm guarantee for the protection of human rights and the protection of persons belonging to minorities. Both sides underline their commitment to the protection of human rights and the protection of persons belonging to minorities."
Estonia's foreign minister, Kristiina Ojuland, told RFE/RL that Estonia is satisfied with the agreement.
"I am very pleased to be able to note that the EU position that was formulated [on 23 April] has remained the same. It has been passed on to the Russian side. This position, in my opinion, embodies fully the solidarity principle inherent in European Union [membership]," Ojuland said.
Irish Foreign Minister Brian Cowen, speaking for the current EU presidency, said he expects the agreement will be signed today.
That sentiment was echoed by the EU's external relations commissioner, Chris Patten.
Russia has been holding out for wording that is more critical of the governments of Estonia and Latvia.
But its suggestion -- that the EU add another sentence supporting the "social integration" of minorities -- was refused following talks yesterday.
An EU source told RFE/RL the sentence says that further support will be provided to the "social inclusion" of the minorities. It is not clear if Russia would accept such wording.
Estonian Foreign Minister Kristiina Ojuland refused to comment on any further possible modifications.
"At this stage, it would not be politic to comment on the text in any greater detail. It meets the Estonian position, and is within the confines of the mandate provided by the Estonian government [on 22 April]. And I think this text is, in any case, a very significant improvement over the draft first tabled by the European Commission a week ago," Ojuland says.
Latvian Foreign Minister Rihards Piks added he was very satisfied with yesterday's compromise, and urged Russia not to press for further modifications.
A number of sources told RFE/RL that Latvia was highly concerned with securing for itself a binding guarantee from the other member states, as well as from the European Commission, that it fully meets the Copenhagen political EU membership criteria.
The issue has received relatively little attention in Latvia, but has become politically very sensitive in Estonia.
Accordingly, Estonia last week listed a number of tightly defined conditions without the fulfillment of which it would be unable to support any reference to its minorities.
Among them were a refusal to accept the naming of specific member states or groups of member states who might suffer from minority issues. Any references to minorities were also to be non-specific. Finally, Estonia said it would reject the term "integration" as too far-reaching.