"I just hope he hasn't gone too far in Speaker Pelosi's direction, and the AFL-CIO's direction that he might lose some support here," he said. "My concern is that what the administration presented has now been moved demonstrably to Democrats, the direction that they wanted."

Cornyn said he expected senators would get a briefing on Wednesday, though other lawmakers warned the date could be in flux until the White House and Democrats finalize any agreement on the U.S.-Canada-Mexico Agreement (USMCA).

"I mean if they got the unions my assumption is that there are going to be a lot of things that were changed from the last time we've seen anything on this. So yeah, I think there's a potential that you could have in their ... desire to try to satisfy Democrats in the House, labor unions cost you some Republican support depending on what those changes are," Thune said.

Pressed if he thought Republicans had been cut out of the process, Thune said some chairmen have been consulted but, "I do think that the play has been for the past several weeks between the House Democrats and the White House."

"Some of the things that we're hearing would be, yeah, would be I think problematic," he added.

A deal on the USMCA would come at a time when House Democrats are moving to impeach Trump and would also benefit Democrats by helping the party argue it is legislating even as impeachment moves forward.

House Democrats have eyed trying to get a vote on USMCA by the end of the year as they barrel toward a jam-packed floor schedule that also includes funding the government and a potential vote on impeachment articles.

Thune warned that House Democrats would have to "hustle" to get a trade deal to the Senate before the holidays, and warned that Senate consideration of USMCA could wait until early 2020.

"My guess is it could end up getting pushed into sometime next year," he said. "I hope that's not true."