US intelligence agencies have assessed that Russia meddled in the 2016 US presidential election by hacking Democrats' emails and distributing online propaganda to help Trump win the election over Democratic rival Hillary Clinton.
Moscow has denied any interference, and Trump has said that his campaign did not collude with Russia.
A Russian presidential envoy said this week that Moscow and Washington were in talks to create a joint cyber security working group.
Asked whether it was a good idea to set up a cyber security cell with the Russians, Rogers told the annual Aspen Security Forum: "I'm not a policy guy here. .... I would argue now is probably not the best time to be doing this."
Senior Republicans criticized the idea, saying Moscow could not be trusted, and Trump later appeared to back away, saying in a post on Twitter: "The fact that President Putin and I discussed a Cyber Security unit doesn't mean I think it can happen. It can't."
Widespread concern and confusion about the possible cyber working group reflects the Trump administration's struggles to articulate a clear approach to international cyber security issues amid inconsistent statements by the president and ongoing personnel changes.
Rogers said such a unit was "something that you might want to build over time were we to see changes in (Russia's) behavior."