Rosenstein also discussed secretly recording the president in May of 2017.
The Times said Rosenstein told Justice Department and FBI officials the secret recordings could be used to expose the chaos of the administration.
It was the most detailed US accusation yet that Russia meddled in the election with the eventual aim of trying to boost Trump's campaign. Russia denies interfering.
Newly reported memo says Trump asked Rosenstein to reference Russia in Comey's firing.
"There are people who have been making threats privately and publicly against me for quite some time," revealed Deputy Attorney General Rod Rosenstein.
Rod Rosenstein rose to fame this week because of a strongly worded letter he wrote about the FBI Chief