TAIPEI, June 18 (Reuters) - Taiwan will allow Terry Gou, the billionaire founder of Taiwan's Foxconn, and TSMC to negotiate on its behalf for COVID-19 vaccines, a government spokesman said on Friday, but warned there was no guarantee of success.

Taiwan is trying to speed up the arrival of the millions of vaccines it has on order as it deals with a rise in domestic cases, although infections remain comparatively low. Only around 6% of Taiwan's 23.5 million people have received at least one of the two-shot coronavirus vaccine regimen.

The government has come under pressure to allow private firms to buy vaccines, and said while it welcomes the idea and is willing to help, they have to provide proof from vaccine producers they have shots and are willing to sell them.

Speaking to reporters after Gou said he was seeking a meeting with President Tsai Ing-wen to discuss his plans to buy 5 million shots from Germany's BioNTech SE, Cabinet spokesman Lo Ping-cheng said Taiwan Semiconductor Manufacturing Co Ltd (TSMC) wanted to donate an equal amount.

Lo said BioNTech had told Gou, who wants to make the purchase through his Yonglin Education Foundation, on Wednesday they will only sell vaccines to governments.

"A good method is for the government to, under this framework, authorise TSMC and the Yonglin Education Foundation to buy them and then donate to the government," he said.

He said the government was, with the help of the German government, continuing its own talks with BioNTech, adding there was no guarantee Gou or anyone else would succeed.

"Even if Mr. Gou can discuss this with the original manufacturer or an agent, can he get them to sell sufficient vaccines? Honestly, nobody knows."

TSMC said it "confirmed the initiative", but offered no other details. BioNTech declined to comment.

Presidential office spokesman Xavier Chang said Tsai had met Gou and TSMC Chairman Mark Liu on Friday to discuss their proposals in a "good and friendly" atmosphere.

"We all hope that it is produced in the original factory with the original packaging and will be delivered directly to Taiwan," he said, referring to the BioNTech shots.

Gou's spokesperson Amanda Liu said after the meeting that a "consensus was reached on key issues", and quoted Gou as saying "It's a daunting mission and there's a long way to go and we will do our best!"

Taiwan's own deal with BioNTech fell through this year, with the government blaming it on pressure from Beijing.

China has denied the accusation, saying Taiwan is free to obtain the vaccines through Shanghai Fosun Pharmaceutical Group Co Ltd, which has a contract with BioNTech to sell the vaccines in China, Hong Kong, Macau and Taiwan.

Taiwan has been dealing directly with BioNTech in Germany, saying it does not trust vaccines from China.

Shieh Jhy-wey, Taiwan's de facto ambassador in Berlin, said that Germany's Economy Ministry had appointed officials to talk directly with BioNTech, adding the company had told him they were willing to sell vaccines to Taiwan directly from Europe.

Germany's Economy Ministry declined to comment.

Japan donated 1.24 million AstraZeneca Plc shots to Taiwan this month and the United States has pledged 750,000 doses, although it has not given details, all on top of vaccines Taiwan has already ordered.

Another 240,000 Moderna Inc doses arrived in Taiwan later on Friday. (Reporting by Ben Blanchard and Yimou Lee; Additional reporting by Beijing newsroom and Andreas Rinke in Berlin; Editing by Raju Gopalakrishnan, Christopher Cushing and Alexander Smith)