By Phil Stewart and Valerie Volcovici

WASHINGTON (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."

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"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.

Those probes are focused almost exclusively on Moscow’s actions, lawmakers and intelligence officials say, and no evidence has surfaced publicly implicating other countries despite Trump's suggestion that others could have been involved.

Moscow has denied any interference, and Trump says his campaign did not collude with Russia.

Representative Adam Schiff, the top Democrat on the House Intelligence Committee, told CNN's "State of the Union" program that Russia could not be a credible partner in a cyber security unit.

"If that’s our best election defense, we might as well just mail our ballot boxes to Moscow," Schiff added.

Separately, U.S. government officials said a recent hack into business systems of U.S. nuclear power and other energy companies was carried out by Russian government hackers, the Washington Post reported on Saturday.

'TIME TO MOVE FORWARD' WITH RUSSIA

Trump said he "strongly pressed President Putin twice about Russian meddling in our election. He vehemently denied it."

He added: "We negotiated a ceasefire in parts of Syria which will save lives. Now it is time to move forward in working constructively with Russia!"

In Trump's first attempt at ending the six-year Syrian civil war, the United States, Russia and Jordan on Friday reached a ceasefire and "de-escalation agreement" for southwestern Syria. The ceasefire was holding hours after it took effect on Sunday, a monitor and two rebel officials said.

Any joint U.S.-Russia cyber initiative would have been a different matter. Depending how much it veered into military or espionage operations, it could have faced major legal hurdles.

Language in the 2017 National Defense Authorization Act prohibits the Pentagon, which includes the National Security Agency and the U.S. military's Cyber Command, from using any funds for bilateral military cooperation with Russia.

Michael McFaul, a former U.S. ambassador to Russia, also noted restrictions on sharing information with Russia that would clearly prohibit offering Moscow a sense of U.S. cyber capabilities. Russia would be similarly adverse to revealing its capabilities to the United States, he noted.

"It just will not happen," McFaul told Reuters.

(Additional reporting by Yasmeen Abutaleb, Jeff Mason and Roberta Rampton; Writing by Arshad Mohammed and Phil Stewart; Editing by James Dalgleish and Peter Cooney)