(Adds comments from Mongolian government, detail)

MELBOURNE/ULAANBAATAR, Nov 25 (Reuters) - Rio Tinto Ltd boss Jakob Stausholm has made his first trip to Mongolia while in the top job to meet Prime Minister Luvsannamsrain Oyun-Erdene as Rio looks to relieve an impasse over the development of a huge copper mine.

Rio is facing costly delays at the underground expansion of its Oyu Tolgoi project that it has blamed on geotechnical difficulties and COVID-19, antagonising the Mongolian government and exacerbating a funding spat with shareholders of mine developer Turquoise Hill.

"This week for the first time since becoming Chief Executive, I was able to return to the beautiful wintry country of Mongolia, home to Oyu Tolgoi, the world's largest greenfield underground copper mining project," Stausholm said in a post on LinkedIn on Wednesday.

As well as meeting with Oyun-Erdene, Stausholm engaged other ministers and members of parliament, "to hear and understand their concerns," he said.

"This was a vital first step in our relationship reset that engages based on trust and we are committed to earning it step by step," he said.

Rio last month delayed first production at the underground development of Oyu Tolgoi by three months to January 2023, after Turquoise had estimated additional funding required for the project had ballooned to $3.6 billion.

Oyun-Erdene stressed Rio should be held liable for the delays in the underground mine development and that it should cover any necessary additional expenditures, waive the debt on the government's 34% stake in the project and ensure that Mongolia receives profits from the project, according to a government press briefing.

Rio Tinto should also terminate immediately the Oyu Tolgoi Underground Mine Development and Financing Plan, the Prime Minister said, according to the briefing.

Mongolia has been lobbying Rio to cancel the deal underpinning the mine's underground expansion, saying rising costs and project delays have eroded expected benefits.

Sources with knowledge of the matter told Reuters earlier this year that Rio was open to replacing the underground development agreement with a new deal ratified by parliament.

Discussions and concessions could include fees Rio gets for managing the project and interest rates on partner loans. (Reporting by Melanie Burton in Melbourne and Anand Tumurtogoo in Ulaanbaatar Editing by Stephen Coates and Mark Potter)