A significant business partner of Michael D. Cohen, President Trump’s personal lawyer, has quietly agreed to cooperate with the government as a potential witness, a development that could be used as leverage to pressure Mr. Cohen to work with the special counsel examining Russian interference in the 2016 presidential election.

Under the agreement, the partner, Evgeny A. Freidman, a Russian immigrant who is known as the Taxi King, will avoid jail time, and will assist government prosecutors in state or federal investigations, according to a person briefed on the matter.