BRUSSELS, March 4 (Reuters) - NATO, that has strained relations with Russia, on Tuesday congratulated Dmitry Medvedev on being elected Russia's next president and said it looked forward to working with him to strengthen cooperation.

"NATO looks forward to working with the new President ... to strengthen our practical cooperation, and to step up our dialogue," NATO Secretary-General Jaap de Hoop Scheffer said in a statement.

Scheffer said he had sent a letter of congratulations to Medvedev.

His statement made no mention of the conduct of Sunday's election, despite concerns expressed in many NATO capitals about curbs that Russia imposed on international observers.

NATO has a special relationship with its old Cold War adversary through a forum called the NATO-Russia Council.

But the relationship is a difficult one with deep Russian unhappiness over the prospect of more former Soviet republics, including Ukraine and Georgia, being allowed to join the U.S.-led Alliance.

Medvedev, elected with 72 percent of the vote, has pledged to pursue the policies of outgoing President Vladimir Putin, who is expected to retain influence as prime minister.

Ukraine has applied to NATO for a Membership Action Plan (MAP), the first step on the path to accession. NATO members are expected to discuss the application, and one from Georgia, at the Bucharest summit.

Putin warned last month Russia could train nuclear missiles on Ukraine if it joined NATO and if this led to Washington stationing elements of a controversial U.S. missile defence shield there.

He issued a similar warning to NATO members Poland and the Czech Republic -- former Soviet satellites states -- last year.

Ukraine's request for a MAP will be discussed by NATO foreign ministers meeting in Brussels on Thursday.

The NATO statement came after U.S. President George W. Bush telephoned Medvedev to say he looked forward to working with the new president and to express the hope of a close working relationship on important world issues. (Reporting by David Brunnstrom; Editing by Richard Balmforth)

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