Speaker Mike Johnson says he speaks with Trump 'frequently' about border negotiations

House Speaker Mike Johnson, a central figure in border negotiations, said he has "frequently" discussed the issue with former President Donald Trump.

Johnson, who is digging in on hard-line demands for more restrictive reforms despite progress in the Senate on a bipartisan compromise, told Fox News that he and Trump talked about the matter on Monday -- before Johnson and other congressional leaders convened at the White House for a meeting with President Joe Biden on Wednesday.

Trump has expressed opposition to any deal between Senate negotiators and the Biden administration.

"President Trump is not wrong," Johnson told Fox News host Laura Ingraham on Wednesday night. "He and I have been talking about this pretty frequently. I talked to him the night before last about the same subject."

Johnson shortly after also appeared on CNN, where he told Kaitlan Collins that he speaks to Trump "every few days."

Trump, in a social media post, singled out Johnson, writing he has "no doubt that our wonderful Speaker of the House ... will only make a deal that is PERFECT ON THE BORDER."

Johnson noted that there is no text out from the Senate on what they've worked out on immigration, and that "we have to reserve judgement to see what comes out of it."

But pouring cold water on the developments, Johnson said on Fox: "It does not sound good at the outset."

Democrats are contending Republicans don't want a border deal before the 2024 election, as it is proving to be a key wedge issue among voters.

Rep. Jamie Raskin, D-Md., said during a House Oversight Committee hearing earlier Wednesday -- before Johnson's comments to Fox and CNN -- that Republicans are "taking orders from Donald Trump and actively obstructing a bipartisan border deal."

Sen. Chris Coons, D-Conn., told CNN on Thursday morning that some House Republicans in recent days have "said they don't want to fix the immigration system in advance of the election."

"My hope is that the cooler heads will prevail in the Republican caucus and those who've worked very hard here in the Senate, both Democrats and Republicans, to give President Biden both the funding he's requested to secure the border and policy changes they say are essential -- that when we pass that in the Senate, the House will recognize this country can't wait 10 months more, this is urgent and we need to make progress on securing our border," Coons said.

Biden invited the "big four" congressional leaders -- Johnson, Senate Majority Leader Chuck Schumer, Senate Minority Leader Mitch McConnell and House Minority Leader Hakeem Jeffries -- to the White House on Wednesday to discuss his stalled supplemental national security package that includes urgent aid for Ukraine (as well as money for Israel and Taiwan) as well as some $14 billion for the border to hire more patrol agents, immigration judges and other resources.

Johnson emerged from the meeting adamant that the border be the first concern.

"We understand that all these things are important, but we must insist, we must insist that the border be the top priority," Johnson said, adding that the "status quo is unacceptable."

Johnson, speaking with Fox's Ingraham, again repeated his call for H.R. 2 -- the House GOP border bill that's been soundly rejected by Senate Democrats and the White House -- or a "functional equivalent." That bill would slash asylum, restart border wall construction and reinstate the Trump-era "Remain in Mexico" policy, among other provisions.

Schumer, speaking on the Senate floor on Thursday, called the White House meeting on the supplemental package constructive but said "it's not a done deal yet."

"We have a number of disagreements we are still working through on issues as complex as immigration and national security," Schumer said. "What matters is not just what we do, but how we do it. The smallest details matter immensely. And it takes time to work through those details. Nevertheless, talks are trending in the right direction. And I remain optimistic we'll get it done soon here in the Senate."

Schumer, without naming any one lawmaker but appearing to address Republicans who've drawn a red line on what they will accept, also said: "Anyone who says 'it's my way or the highway' on border is not serious about reaching an agreement. It must be bipartisan."

ABC News' Mariam Khan contributed to this report.