Washington (CNN) Senate Minority Leader Chuck Schumer said Wednesday talks on immigration and border security are "starting over" after he and the White House have exchanged a series of blows about President Donald Trump's border wall.

But even as the New York Democrat described a reset in talks, Senate Majority Whip John Cornyn told reporters that he'd propose something similar to what Schumer was talking about moving on from.

"We're starting over," Schumer told CNN on Wednesday when asked about the latest on the standoff over the Congress' plans for addressing the expiring Deferred Action for Childhood Arrivals policy . "I took our thing off — they took their thing off the table, I took our thing, we're starting over."

Schumer was referring to an offer he made Trump last week to authorize upward of $20 billion for his border wall, a signature campaign pledge for the President, in exchange for protecting recipients of DACA, young undocumented immigrants who came to the US as children. Trump decided last fall to end the program by March 5, 2018, and Congress has since failed to reach agreement with the White House about how to extend it.

But the White House rejected Schumer's offer and after the government reopened from a weekend shutdown, Schumer rescinded the offer . That prompted Trump to jab back Tuesday night.

"Cryin' Chuck Schumer fully understands, especially after his humiliating defeat, that if there is no Wall, there is no DACA. We must have safety and security, together with a strong Military, for our great people!" Trump tweeted

A cheery Schumer nonetheless told CNN on Wednesday that work would continue, and as he entered an elevator with Sen. Lamar Alexander, he coaxed a fist-bump from the Tennessee Republican who has been working with a bipartisan group of senators to find common ground on immigration and other issues.

"We're trying to make (Congress) function," Schumer said, indicating Alexander, ignoring a question about whether he would meet with Alexander's group.

A bipartisan group will meet on immigration at 5 p.m. Wednesday and has grown to more than 40 lawmakers, aides say. Unlike previous meetings, the large group will also include senators considered more hardline on both sides.

Cornyn proposes 'permanent for permanent' framework

Earlier Wednesday, however, Cornyn told reporters that he would support a trade of border security for a DACA-type solution, putting him at potential odds with the White House.

The Texas Republican called his offer "temporary for temporary, permanent for permanent." In essence, he said, if lawmakers want a "permanent" solution on DACA -- a pathway to citizenship for the eligible immigrants -- then they need to be prepared to pony up for a "permanent" border fix. And the inverse is also true.

"If you want an annual appropriation, then I think you'll get a one-year extension of the DACA status," Cornyn said. "If you want a permanent solution for the DACA recipients, you're going to need a permanent solution -- which means a plan and funding, something on the order of what Sen. Schumer initially offered, $25 billion, to the President last Friday, which he has now rescinded."

Cornyn said funding in that range, which would cover 10 years, would need to be put together all up front, and likely put into a trust fund that can then be used flexibly by the Department of Homeland Security for what they need -- infrastructure and wall, technology or personnel.

"I'd leave it to the experts to say what works best at any given location," Cornyn said, adding later, "Different places will have different requirements, so I'd leave it flexible for the Department of Homeland Security."

He said unlike a DHS proposal that was recently sent to the Hill and unlike a bipartisan proposal Trump rejected, Cornyn would not accept appropriating funds year-by-year.

"I disagree with that approach," Cornyn said.

But Cornyn acknowledged his framework may not speak for everyone, including the White House, which has pushed for a DACA deal to include sweeping changes to the immigration system like slashing family-based and diversity visas and more aggressive enforcement authorities.

"Not everybody sees it the same way," Cornyn said.

Cornyn said senators have agreed to negotiate further and clear ideas through himself on the Republican side and Senate No. 2 Democrat Dick Durbin on the other.