In July, Trump said in an interview with the New York Times that if Mueller examined his family's finances beyond any relationship with Russian Federation he'd consider it "a violation".
Both reports said Mueller issued the subpoena several weeks ago.
Deutsche Bank, which has loaned the Trump organisation millions of dollars for real estate ventures, said it would not comment on any of its clients.
Mr Mueller is investigating claims of collusion between Trump's aides and Russian officials while also probing Moscow's influence in the United States presidential election which brought Trump, the billionaire businessman into power. It is not clear whether Mueller is interested in the bank accounts because they are connected to the Russian Federation probe or if he is investigating another matter.
Mueller's probe, established on May 17, is mandated to continue the FBI investigation into the Russian government's alleged efforts to interfere with the 2016 USA presidential election.
While Trump campaigned during the 2016 election, the bank rejected calls to be transparent about exactly how much money Trump owes and why.
As Mueller's investigation unfolds, Trump has gone on the offensive.
Arrest made after missing North Carolina girl presumed dead
He is charged with concealing of death, obstruction, second-degree burglary, felony larceny and possession of stolen property. Highway 17 or state Highway 53 in the area where the girl disappeared to call police and let them see the footage.
"I think that's a violation", Mr. Trump told the paper.
CNN Congressional Correspondent Manu Raju tweeted that Republicans on the House Intelligence Committee actually blocked the Trump finances subpoena, refusing to order Deutsche Bank to hand over the Trump Organization's financial records.
On Friday, Mueller revealed a guilty plea by Michael Flynn, Trump's former national security adviser, who admitted lying to Federal Bureau of Investigation agents.
The Trump International Hotel is shown on August 10, 2017 in Washington.
In October, Manafort pleaded not guilty to charges including conspiracy to launder money and conspiracy against the United States.
The special counsel has over a dozen lawyers working for his office.
In May, Deputy Attorney General Rod Rosenstein appointed Mueller to serve as special counsel for the Justice Department, and to oversee the investigation into "any links and/or coordination between Russian government and individuals associated with the campaign of President Donald Trump, and any matters that arose or may arise directly from the investigation".