Mr. Mueller could run afoul of a line the president has warned him not to cross. Though it is not clear how much of the subpoena is related to Mr. Trump’s business outside ties to Russia, Mr. Trump said in an interview with The New York Times in July that the special counsel would be crossing a red line if he looked into his family’s finances beyond any relationship with Russia. The president declined to say how he would respond if he concluded that the special counsel had crossed that line.

Mr. Trump campaigned as a businessman whose deal-making prowess would translate directly into reforming Washington. The argument helped propel him to the White House, but the Trump Organization has been a magnet for criticism from Democrats, ethics watchdogs and some Republicans, who expressed concern that he remained vulnerable to conflicts of interest because he did not separate from the company.

Before Mr. Trump was sworn in, he pledged that he would stay uninvolved in his businesses while in office but insisted it would be too punitive for his business partners for him to divest from the company altogether.

Among the Trump Organization’s holdings are golf clubs, hotels and licensing agreements for the use of the Trump name on properties and other products. While its holdings are complex, the company has always been run like a small, family-owned business; Mr. Trump brought in his three eldest children to help run the enterprise.

The Trump Organization is not publicly held, making it difficult to determine where it receives its money and invests it. The company has said that it never had real estate holdings in Russia, but witnesses recently interviewed by Mr. Mueller have been asked about a possible real estate deal in Moscow.

In 2015, a longtime business associate of Mr. Trump’s, Felix Sater, emailed Mr. Trump’s lawyer, Michael Cohen, at his Trump Organization account claiming he had ties to President Vladimir V. Putin of Russia and that building a Trump Tower in Moscow would help Mr. Trump’s presidential campaign. Mr. Trump signed a nonbinding letter of intent for the project in 2015 and discussed it at least three times with Mr. Cohen.

A revealing comment about Russia by Eric Trump, the president’s middle son, also drew scrutiny when it emerged last year. James Dodson, a longtime golf writer from North Carolina, said offhand in a radio interview that Eric Trump, who oversees the golf courses for the Trump Organization, told him in 2013 that the Trumps relied on Russian investors to back their golf clubs. Eric Trump has denied those remarks.