(Recasts with Trump statement, reaction)

By Phil Stewart and Valerie Volcovici

WASHINGTON, July 9 (Reuters) - U.S. President Donald Trump on Sunday backtracked on his push for a cyber security unit with Russia, tweeting that he did not think it could happen, hours after his proposal was harshly criticized by Republicans who said Moscow could not be trusted.

Trump said on Twitter early on Sunday that he and Russian President Vladimir Putin discussed on Friday forming "an impenetrable Cyber Security unit" to address issues like the risk of cyber meddling in elections.

The idea appeared to be a political non-starter. It was immediately scorned by several of Trump's fellow Republicans, who questioned why the United States would work with Russia after Moscow's alleged meddling in the 2016 U.S. election.

"It's not the dumbest idea I have ever heard but it's pretty close," Senator Lindsey Graham of South Carolina told NBC's "Meet the Press" program.

Ash Carter, who was U.S. defense secretary until the end of former Democratic President Barack Obama's administration in January, told CNN flatly: "This is like the guy who robbed your house proposing a working group on burglary."

Trump's advisers, including Secretary of State Rex Tillerson and Treasury Secretary Steve Mnuchin, had recently sought to explain Trump's cyber push.

Mnuchin said on Saturday that Trump and Putin had agreed to create "a cyber unit to make sure that there was absolutely no interference whatsoever, that they would work on cyber security together."

But Trump returned to Twitter on Sunday to play down the idea, which arose at his talks with Putin at a summit of the Group of 20 nations in Hamburg, Germany.

"The fact that President Putin and I discussed a Cyber Security unit doesn't mean I think it can happen. It can't," Trump said on Twitter.

He then noted that an agreement with Russia for a ceasefire in Syria "can & did" happen.

Republican Senator John McCain of Arizona acknowledged Trump's desire to move forward with Russia, but added: "There has to be a price to pay."

"There has been no penalty," McCain, who chairs the Senate Armed Services Committee, told CBS' "Face the Nation" program according to a CBS transcript. "Vladimir Putin ... got away with literally trying to change the outcome ... of our election."

Trump argued for a rapprochement with Moscow in his campaign but has been unable to deliver because his administration has been dogged by investigations into the allegations of Russian interference in the election and ties with his campaign.

Special Counsel Robert Mueller is investigating the matter, including whether there may have been any collusion on the part of Trump campaign officials, as are congressional committees including both the House of Representatives and Senate intelligence panels.