VILNIUS, July 10 (Reuters) - Turkish President Tayyip Erdogan on Monday agreed to forward to parliament Sweden's bid to join the NATO military alliance, appearing to end months of drama over an issue that had strained the bloc as war has raged in Ukraine.

Sweden and Finland applied for NATO membership last year, abandoning their policies of military non-alignment that had lasted through the decades of the Cold War in response to Russia's invasion of Ukraine.

While Finland's NATO membership was green-lighted in April, Turkey and Hungary have yet to clear Sweden's bid. Stockholm has been working to join the bloc at the alliance's summit in the Lithuanian capital Vilnius, which begins on Tuesday.

"I'm glad to announce ... that President Erdogan has agreed to forward the accession protocol for Sweden to the grand national assembly as soon as possible, and work closely with the assembly to ensure ratification," NATO Secretary General Jens Stoltenberg told a news conference, describing it as a "historic" step.

He had convened Erdogan and Swedish Prime Minister Ulf Kristersson for several hours of talks on the eve of the summit as he sought to finally break the deadlock.

Erdogan has held out for months, saying Sweden's accession hinged on the implementation of a deal reached last year during the alliance's summit in Madrid and that no one should expect compromises from Ankara.

Turkey has accused Sweden of not doing enough against people Turkey sees as terrorists, mainly members of the outlawed Kurdistan Workers Party (PKK) that is considered a terrorist organisation by Turkey, the EU and the United States.

Erdogan and Kristersson appeared relaxed ahead of the meeting, with the Swedish leader joking about parking his plane next to the bigger Turkish aircraft at Vilnius airport.

"This has been a good day for Sweden," Kristersson told reporters, saying the joint statement on Monday represented "a very big step" toward the final ratification of Sweden's membership of NATO.

The statement issued by both countries said Sweden had reiterated that it would not provide support to the Kurdish groups and would actively support efforts to reinvigorate Turkey's EU accession process.

Erdogan on Monday said the European Union should open the way for Ankara's accession to that bloc before Turkey's parliament approved Sweden's bid to join the NATO military alliance.

Stoltenberg said Erdogan had agreed to push ratification in parliament "as soon as possible," but he could not give a specific calendar. It took two weeks for Turkey's parliament to ratify Finland's membership.

After Hungarian Prime Minister Viktor Orban's chief of staff said on Thursday that Budapest would now no longer block Sweden's NATO membership ratification, Turkish approval would remove the last hurdle for Swedish accession to NATO, applications for which must be approved by all members.

The United States and its allies have sought to pressure Ankara for months. Some NATO partners believe that Turkey, which requested in October 2021 to buy $20 billion of Lockheed Martin Corp (LMT.N) F-16 fighters and nearly 80 modernization kits for its existing warplanes, has been using Swedish membership to pressure Washington on the warplanes.

U.S. President Joe Biden, who welcomed the announcement, is due to hold face-to-face talks with Erdogan during the summit.

Additional reporting by Niklas Pollard, Andrius Sytas and Justyna Pawlak in Vilnius, Anna Ringstrom and Johan Ahlander in Stockholm, and Ezgi Erkoyun in Istanbul; writing by John Irish; editing by William Maclean and Rosalba O'Brien.

By Sabine Siebold and John Irish