The White House denied reports that the US has threatened or is considering any physical threat to Snowden, 29, who is currently in Russia and has sought asylum in Ecuador, which is said to be his final destination from Moscow.
Trump said military action against a defiant North Korea is certainly an option, but hoped that he does not have to use it.
The President has maintained that he has nothing to do with Russia.
A chilling video of beheading of another American journalist, Steven Sotloff, was on Tuesday posted on social media by Syria and Iraq-based terrorist group, Islamic State.
The bill now heads to the White House for President Donald Trump's approval.
Trump said that America's relationship with Russia 'may be at an all-time low', but he hoped that it would have been wonderful if the two countries could get along.