Updated at 9:02 a.m. ET

President Trump says he "strongly pressed" Russian President Vladimir Putin twice about Russia's meddling in the U.S. election — and that it's now time "to move forward in working constructively with Russia."

That work, Trump said, will include a new ceasefire in Syria, as well as a joint cybersecurity operation. But by Sunday evening he distanced himself from the latter, tweeting, "The fact that President Putin and I discussed a Cyber Security unit doesn't mean I think it can happen."

In a series of tweets Sunday about his meetings with Putin at the G-20 Summit, Trump said Putin "vehemently denied" interfering in the U.S. election.

"I've already given my opinion," Trump added, apparently referring to his remarks in Poland shortly before attending the G-20 Summit in Germany. When he was asked about election hacking, Trump replied, "Well, I think it was Russia, and I think it could have been other people in other countries. Could have been a lot of people interfered."

He added, "Nobody really knows for sure."

At that news conference in Warsaw, Trump also noted that "three or four" U.S. intelligence agencies, rather than all of them, say that Russia sought to interfere in the election. And he compared their findings about Russia's involvement to the hunt for weapons of mass destruction in Iraq — making it clear that he remains a skeptic.

Trump's posts on Sunday morning also included digs at the CIA, FBI and the Democratic National Committee, as well as a plan to work with Russia to form a "Cyber Security unit," raising questions about whether America's president agrees with U.S. intelligence agencies' conclusion that Russia carried out hacking to help Trump win the White House.

After Trump touted the plan to collaborate with Russia on Internet security issues, he was criticized by fellow Republicans, including his former rivals for the presidency.

"It's not the dumbest idea I've ever heard, but it's pretty close," Sen. Lindsey Graham, R-S.C., said on NBC's Meet the Press.

Graham called the meeting with Putin "disastrous," and said that while he sees Trump having success in other foreign policy areas, "when it comes to Russia, he's got a blind spot."

Sen. Marco Rubio, R-Fla., had a similar take, saying that "Partnering with Putin on a 'Cyber Security Unit' is akin to partnering with [Syrian President Bashar] Assad on a 'Chemical Weapons Unit.'"

Treasury Secretary Steve Mnuchin, defended the president, asking on ABC's This Week, "Why would President Trump broadcast exactly what he said" to Putin and adding, "Strategically that makes no sense."

It's the second time in recent days Trump has tweeted about the Russian hacking. On Friday, he wrote from the international talks in Hamburg to say, "Everyone here is talking about why John Podesta refused to give the DNC server to the FBI and the CIA. Disgraceful!"

Responding to that post, Podesta clarified that he had worked for Hillary Clinton's presidential campaign and not the DNC. He also reminded the president of his duties at the G-20 conference and told Trump, "Get a grip, man."

At the Hamburg meetings, Trump and Putin spoke for more than two hours. Afterwards, Putin said he thought Trump had "agreed" with his denials of Russia's involvement in the U.S. presidential campaign.

Secretary of State Rex Tillerson has offered a nuanced take on the two leaders' meeting, saying that while Trump raised the "concerns of the American people regarding Russian interference in the 2016 election" at the very start of his time with Putin, Trump also decided to focus on advancing talks on other issues. Tillerson added that in the meeting, "there was not a lot of re-litigating of the past."

When reporters asked Tillerson about how direct the American president had been in confronting Putin over Russia's interference in the U.S. democratic process — and whether Trump had presented any evidence to try to convince Putin — Tillerson replied: