WASHINGTON—A Treasury Department unit that specializes in combating money-laundering will share financial records with an expanding Senate probe into possible ties between Russia and President Donald Trump and his associates, according to people familiar with the matter.

The Senate Intelligence Committee requested the records from Treasury’s Financial Crimes Enforcement Network, or FinCEN, late last month, these people said. The people familiar with the matter didn’t specify the nature of those records. One person said that...