Mr Snowden is technically a guest of the Russian government and in January a spokeswoman for the Russian foreign ministry said he would be allowed to stay for “a couple more years”.
He first arrived in Russia in June 2013 after fleeing from Hong Kong.
Russia does not have an extradition treaty with the US so it is not clear what mechanism would be used to hand Mr Snowden over, though the two countries have exchanged prisoners and spies in the past.
“I am confident that President Putin is creative enough to find a way within his own legal system to deport Snowden,” said Mr Morrell.
Maria Zakharova, a spokeswoman for the Russian foreign ministry, put out a cryptic statement in response to the report, saying that Russia was still formulating its policy towards the US and that Mr Trump’s efforts were being hampered by domestic political opponents.
CNN reported that US intelligence had corroborated parts of a 35-page dossier that former MI6 spy Christopher Steele had created on Mr Trump’s background.
None of the confirmed information relates to salacious allegations about prostitutes that were contained in the dossier, CNN said, and were focused instead on claims about conversations between Russian citizens which the US now believes to be accurate.
Sean Spicer, the White House press secretary, said in response: “We continue to be disgusted by CNN’s fake news reporting.”