There is in fact a clear record of strong cooperation between the EU and Russia, dating back to 1994, when the two sides negotiated a Partnership and Cooperation Agreement. Russia was identified as a strategic partner for the EU, and was the only country with which the EU held summits not once, but twice a year.
In 2010 the EU and Russia launched a Partnership for Modernisation. As stated in the Joint Declaration: "The European Union and Russia, as long-standing strategic partners in a changing multipolar world, are committed to working together to address common challenges with a balanced and result-oriented approach, based on democracy and the rule of law, both at the national and international level" http://bit.ly/1p648gA.
Despite current difficulties in the relationship with Russia, the EU and its Member States have maintained a clear policy of reaching out to Russian society and youth, mainly through the Erasmus+ student exchange programme and other people to people contacts.
NATO has also consistently worked hard to build a cooperative relationship with Russia since the early 1990s: no other NATO partner has been offered a comparable relationship, nor a similarly comprehensive institutional framework: http://bit.ly/2e0TZnG.