Mr Mueller is looking into claims of Russian interference in the U.S. presidential election and potential collusion by Mr Trump's team - an allegation denied by the President and Russian authorities.
Bankruptcies at his hotel and casino business in the 1990s made much of Wall Street shy away from lending Mr Trump credit, forcing him to look further afield.
Trump, his wife, daughter Ivanka and son-in-law Jared Kushner are understood to beclients of Deutsche Bank.
Asked if delving into his and his family's finances unrelated to the Russian Federation probe would cross a red line, Trump replied, "I would say yeah".
Trump has consistently denied any collusion between his campaign and Russian Federation and has stated that he did not have any business dealings in Russian Federation.
Holding such debt, particularly if some of it was or is coming due, could potentially give Russian banks some leverage over Trump, especially if they are state-owned, said a second USA official familiar with Russian intelligence methods. Sources familiar with Trump's borrowing from the bank told The Financial Times the total he owed is more like $300 million. But the fact that Mueller is looking at Deutsche Bank means his probe is reaching the very heart of Trump's financial life.
The president declined to say what he would do if Mueller began such an examination, but he continued, "I think that's a violation".
The Trump International Hotel is shown on August 10, 2017 in Washington.
In June, Deutsche Bank already rejected demands by US House Democrats to provide details of Trump's finances, saying sharing client data would be illegal unless it received a formal request to do so. By sheer coincidence, Deutsche Bank recently had to pay $670 million in fines for its role in a Russian money laundering plot (though as Bloomberg reports, the bank's internal investigation found no link between Trump and that scheme).
Special counsel Robert Mueller's "Russiagate" probe cost United States taxpayers $3.2 million in the first six months, mostly on salaries and benefits.
Since the probe began in May, the special counsel has charged four people, two of whom have pleaded guilty.
In January, Deutsche Bank agreed to pay $630 million in fines for organizing $10 billion in sham trades that could have been used to launder money out of Russian Federation.