Morgan Stanley’s wealth-management arm is being scrutinized by the Federal Reserve, which is looking into whether the bank has sufficient controls in place to prevent rich foreign customers from laundering money. 

The Fed has been probing how Morgan Stanley vets foreigners and the origins of their money before taking them on as customers, according to people familiar with the matter. What started out as a routine review a few years ago escalated after regulators found the bank’s due diligence on clients and anti-money-laundering efforts lacking.