STANSTED, July 9 (Reuters) - U.S. President Joe Biden arrived in Britain on Sunday, starting a three-nation trip that will be dominated by a NATO summit in Lithuania aimed at showing solidarity with Ukraine in its fight against Russia while not yet accepting Kyiv as a member of the alliance.

But the challenges of forging solidarity among NATO's 31 member countries were highlighted in a call between Biden and Turkish President Tayyip Erdogan before the alliance summit in Lithuania this week, with Sweden's bid for membership in the Western alliance a continued point of contention.

Biden landed at Stansted Airport and boarded the Marine One helicopter for central London, where he will meet British Prime Minister Rishi Sunak at 10 Downing Street on Monday. He will later travel to Windsor Castle for a visit with King Charles.

The talks with the king, expected to include climate initiatives, will give Biden a greater sense of the man who succeeded his mother, Queen Elizabeth, after her death last September.

Biden had tea with the queen at Windsor in June 2021, and they discussed many of the same issues that remain a top priority today, like Russia and China.

Biden will travel on to Vilnius, Lithuania, on Monday night and hold talks with NATO leaders there on Tuesday and Wednesday. Biden and the NATO allies aim to show support for Ukraine and give Ukraine President Volodymyr Zelenskiy a sense of what will have to be done to gain NATO membership sometime in the future.

In a CNN interview previewing his trip, Biden urged caution for now on Ukraine's drive to join NATO, saying the alliance could get drawn into the war with Russia due to NATO's mutual defense pact.

"I don't think there is unanimity in NATO about whether or not to bring Ukraine into the NATO family now, at this moment, in the middle of a war," Biden said.

Zelenskiy said an invitation for Ukraine to join NATO would send a message that the Western defense alliance is not afraid of Moscow. Ukraine should get clear security guarantees while it is not in NATO, and Zelenskiy said that would be one of his goals in Vilnius, in an interview broadcast on Sunday.

"I'll be there and I'll be doing whatever I can in order to, so to speak, expedite that solution, to have an agreement with our partners," Zelenskiy said on ABC's "This Week."

The NATO membership of Sweden, whose accession to the alliance has been blocked by both Hungary and Turkey, will be part of the agenda in Vilnius. New members must be approved by a unanimous vote of all existing NATO members.

Biden discussed Sweden's NATO bid on a call with Erdogan, and "conveyed his desire to welcome Sweden into NATO as soon as possible," the White House said in a statement on Sunday.

Erdogan told Biden that Sweden must do more to contain supporters of the Kurdistan Workers Party (PKK), which it considers a terrorist group and who continue to hold demonstrations in Sweden, Erdogan's office said.