The inquiry focuses on whether people from Middle Eastern nations — including Qatar, Saudi Arabia and the United Arab Emirates — used straw donors to disguise their donations to the two funds. Federal law prohibits foreign contributions to federal campaigns, political action committees and inaugural funds.

The line of questioning underscores the growing scope of criminal inquiries that pose a threat to Mr. Trump’s presidency. The special counsel, Robert S. Mueller III, is focusing on whether anyone in the Trump campaign conspired with Russia to tip the 2016 presidential election in Mr. Trump’s favor, while prosecutors in New York are pursuing evidence he secretly authorized illegal payments of hush money to silence accusations of extramarital affairs that threatened his campaign.

The inquiry into potential foreign donations to the inaugural fund and the super PAC is yet another front being pursued by multiple teams of prosecutors. Thomas J. Barrack Jr., a billionaire financier and one of Mr. Trump’s closest friends, raised money for both funds.