They say the rejection of the recommendation was standard bank procedure when lucrative clients were involved.
"The Failing New York Times (it will pass away when I leave office in 6 years), and others of the Fake News Media, keep writing phony stories about how I didn't use many banks because they didn't want to do business with me", Trump said in the first of a series of tweets.
"At no time was an investigator prevented from escalating activity identified as potentially suspicious" to the Treasury Department, a spokesman for the bank told AFP news agency.
Trump has a long history of partnership with Deutsche Bank, which is believed to have lent him around $2bn (£1.6bn), with hundreds of millions still owed to the German bank.
The report didn't give specifics about the nature of the transactions but said some of them involved money being transferred back and forth with foreign entities or individuals.
Former employee Tammy McFadden said she was terminated a year ago after raising concerns about the bank's practices. "It is because I didn't need money", Trump tweeted.
The Times explained that certain transactions, particularly involving real-estate, can sometimes automatically trigger review procedures without any wrongdoing. The bank has been accused of laundering billions for the Kremlin and other Russian agencies, which has fueled suspicions the bank may have been a conduit between Trump and Moscow.
The newspaper said a Kushner Cos spokeswoman called any allegations of relationships involving money laundering "made up and totally false". Banks have always been available to me, they want to make money.
In response to his tweets he buys everything with cash, Trump added: "Where did he get all of that cash?" The president said on Twitter that he didn't need other banks and that Deutsche Bank "was very good and highly professional to deal with - and if for any reason I didn't like them, I would have gone elsewhere".
Congressional and NY state authorities are investigating the relationship between Trump, his family and Deutsche Bank, and demanding documents related to any suspicious activity.
As the Times reported in March, "Deutsche Bank officials have quietly argued to regulators, lawmakers and journalists that Mr. Trump was not a priority for the bank or its senior leaders and that the lending was the work of a single, obscure division".
On April 30, Trump and three of his children sued two banks - Germany's Deutsche Bank and US-based Capital One - to block them from releasing financial records to the Congress.
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