Before President Trump headed to meet Vladimir Putin last month, Deputy Attorney General Rod Rosenstein requested a meeting in the Oval Office. He was ready to indict Russian officials for election hacking and wanted to know if the president wanted the Justice Department to announce the charges before or after the trip.

Mr. Trump told Mr. Rosenstein to issue the statement as soon as possible, adding that it would strengthen his position in talks with Moscow, according to people familiar with the exchange.

The...