By Jonathan Allen

NEW YORK (Reuters) -- The U.S. Republican Party filed a formal complaint against one of Hillary Clinton's family charities with the Internal Revenue Service on Tuesday and is calling for an audit after the charity said this week it would not refile erroneous tax returns.

The Bill, Hillary & Chelsea Clinton Foundation's flagship health program conceded earlier this year it had misreported by millions of the dollars the money it was given by governments compared with non-government donors in its tax returns for 2012 and 2013.

The charities have come under intense scrutiny this year with Hillary Clinton remaining the favorite to become the Democratic Party's nominee in the November 2016 presidential election.

The Clinton Health Access Initiative (CHAI) originally said it was amending the forms, known as form 990s, after Reuters discovered the errors in the spring. This week, CHAI said it had decided against refiling, saying the errors had "no material impact," CHAI spokeswoman Maura Daley said.

The Republican National Committee disagrees, its chairman, Reince Priebus, said on Tuesday in his letter to IRS Commissioner John Koskinen, a copy of which was given to Reuters.

"The American people deserve to know whether the largest philanthropic arm of the Clinton Foundation continues to misreport the funds it receives from foreign governments, and whether this might lead to the potential for further conflicts of interest," Priebus wrote.

Daley declined to comment beyond repeating that CHAI "does not believe a refiling is necessary." CHAI says the total amount of income was correctly reported for both years, but the breakdown of government against non-government funding had been muddled up.

Spokesmen for Hillary Clinton did not respond to requests for comment.

Republicans have said the charities, which rely heavily on foreign government funding for their work on health, educational and environmental projects, present a hopeless conflict of interest for Clinton, who has dismissed this as a baseless political attack. She severed her formal ties with the charities after announcing her candidacy in April, but her husband and daughter continue to have leading roles.

The criticism intensified in April after the charities conceded that they did not fully comply with the ethics agreement Clinton signed with Barack Obama's incoming presidential administration in 2008 in order for her to become his secretary of state.

A spokesman for the IRS said charities should refile a form 990 if it becomes aware of an error. Federal law bans the IRS from discussing specific cases, and it remains unclear whether it will agree that an audit of CHAI is necessary.

While it is not unusual for charities to refile a form 990 on occasion and that doing so is not evidence of wrongdoing, several tax experts told Reuters it was odd for a charity to make large mistakes several years in row. CHAI had previously refiled its 2010 and 2011 tax returns after over-reporting the amount it received from government grants by tens of millions of dollars.

The form 990s are used by the IRS to ensure a charity is not in breach of its tax-exempt status, and the forms must be made public so people can see how a charity raises and spends money.

The Clinton Foundation, which files tax returns separately from CHAI, also said earlier this year it expected to refile three erroneous tax returns after Reuters found it had wrongly reported receiving nothing from foreign governments in 2010, 2011 and 2012. A foundation spokesman told news website Politico, which first reported CHAI's decision not to refile, this week that it was still reviewing its accounts for those years.

(Reporting by Jonathan Allen; Editing by Lisa Shumaker)