The 1996 accord has led to closer coordination on travel visas, the environment, and in fighting illegal immigration, but has been criticized for not doing enough to promote human rights and democratic reforms.
The new comprehensive agreement would cover trade, human rights, and the rule of law, as well as energy cooperation.
The European Commission, the EU's executive, wants member states to agree on the draft in time for an EU-Russia summit in November. However, reports say rifts remain unresolved between former Eastern European countries that regard Moscow with suspicion and Western European states that see it as a reliable energy provider.
EU External Relations Commissioner Benita Ferrero-Waldner said the EU wants a relationship based "on the same principles and rules."