TAIPEI, May 4 (Reuters) - A former U.S. ambassador appointed by Donald Trump said on Saturday he believed the former president would again support Taiwan if he wins back the White House.

Taiwan, claimed by China as its own territory despite Taipei's objections, received strong backing from Trump's 2017-2021 administration, including arms sales, which have continued under the government of President Joe Biden.

Trump spoke to Taiwan President Tsai Ing-wen in 2016 shortly after he won the election, prompting anger in Beijing - as the United States does not officially recognise Taiwan's government - and glee in Taipei.

James Gilmore, Trump's ambassador to the Organization for Security and Cooperation in Europe, told reporters in Taipei that Trump was not an isolationist but was trying to get U.S. allies to take their own defence more seriously.

"I believe that President Trump will be supportive of Taiwan when he becomes president. He was in his first term," said Gilmore, visiting Taipei to speak at the Taiwanese think tank the Center for Asia-Pacific Resilience and Innovation.

Trump, the presumptive Republican nominee, is set to face Biden, a Democrat, in a rematch in November's presidential election.

Gilmore will meet Tsai on his trip though not President-elect Lai Ching-te, who takes office on May 20, due to scheduling issues.

He said he would report back to Trump on his visit, and pass on any messages from Taiwan if given them.

"I fully expect to write a memorandum and submit it to President Trump. What he does with these memos people send him we do not know," he said. "But I have made up my mind that I can be helpful."

Gilmore, the Republican governor of Virginia from 1998 to 2002, added he thought lines of communication between Taiwan and the United States were already strong and he did not think he needed to serve as a messenger.

The United States is democratically governed Taiwan's most important international backer and arms supplier, to the frequent anger of China, which has ramped up military and political pressure against the island.

Taiwan has hosted several Republican lawmakers this year as part of bipartisan delegations visiting the island, including in February Mike Gallagher, then-chair of the House of Representatives select committee on China.

Gallagher said on that trip that no matter who wins the elections, the U.S. would continue to support Taiwan.