Rio Tinto operates what is slated to become the world's third-largest copper mine which is jointly owned by the Mongolian government and Canada's Turquoise Hill Resources Ltd, in which Rio owns a 50.8% stake.
Activist investor Pentwater Capital Management LP holds a 9.23% stake in Turquoise Hill and is its largest minority investor.
In a letter to the Australian exchange on Monday, Pentwater Chief Executive Matthew C. Halbower accused Rio of mismanaging costs, and then unfairly restricting Turquoise Hill's ability to obtain finance to pay for those costs.
Rio Tinto declined to comment.
"It is unacceptable that Rio ... is preventing TRQ from seeking the financing solutions that are most optimal for TRQ shareholders," Halbower said.
"If Rio's Board continues with this oppressive behaviour ... Pentwater is prepared to go forward with legal action against the proper parties, including, you, the Board of Directors."
Rio in 2019 announced a cost overrun at the project of up to $1.9 billion due to difficult geology, saying total capital expenditure was expected to be in a range of $6.5 billion-$7.2 billion, and it expected a delay of up to 30 months at the mine's underground extension.
In September, it said it planned to raise up to $500 million through additional lending to develop the mine.
However shareholders worried that a rights issue would allow Rio Tinto to take a greater stake in Turquise Hill that underplayed its full valuation.
More advantageous financing operations included streaming and bond financing, said Halbower who also noted that he stood for a board role but said that was vetoed by Rio.
(Reporting by Melanie Burton; Editing by Stephen Coates)