"I am skeptical that these efforts are going to bear fruit. I think it's very hard to negotiate with this White House when the president's public positions seem to change by the day," Murphy said during a press conference in Connecticut.

Murphy added that he thought "the chances of success here are less than 50-50, but I think it's worth giving this a shot."

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Democrats and outside groups have clamored for new background check legislation in the wake of the mass shootings in Dayton, Ohio, and El Paso, Texas. But Trump has yo-yoed in his public comments about what he would support, giving Republicans cover to avoid taking a stance on specific pieces of legislation or proposals.

Murphy — who was elected to the Senate roughly a month before the 2012 Newtown, Conn., mass shooting — appeared to try to strike a realistic tone about the chances of an agreement.

He described the White House as showing a willingness to negotiate, but noted that Trump had previously suggested he was open to expanding background checks only to back down amid pressure from his base and the gun lobby.

"The president and the White House has been clear that they are willing to support background checks legislation that might not today be popular within the Republican Party. … Again, that's what the president said after Parkland, and he ended up doing the opposite of leading," Murphy said, referring to Parkland, Fla., the site of a previous mass shooting.

He added that he has "made it clear to the White House that if they're prepared to give the [National Rifle Association] NRA veto power over this package then it's not worth having even one meeting."

Murphy has been in talks with the White House for roughly two weeks and noted on Thursday that he believed the White House was still interested in background checks legislation. Staff-level meetings and conversations between senators and the White House are expected to continue while the Senate is out of town until Sept. 8.

"I believe that the White House is still committed to trying to work on a comprehensive anti-gun violence proposal that would include strengthening background checks. And I committed to the White House last night that I would work with them to try to find common ground on expanding background checks in this country," he said on Friday.

He added that the White House also wants a deal on so-called red flag legislation, which would incentivize states to enact laws that make it easier for law enforcement to identify potentially dangerous individuals and temporarily block them from owning or buying a gun.

He noted that Trump's support could result in a background checks bill passing the Senate, but characterized "red flag" legislation as the more likely outcome.

"I do think that that has a better chance of passing. I think there's a lot of uncertainty as to what that is yet," he said. "My worry about extreme risk protection orders is that if all it is is incentives for the states, then are those incentives big enough to convince a Republican governor and a Republican state legislature to adopt a law that the NRA opposes?"