The president said that he could support the deal, but argued an executive order he issued last week designed to change insurance markets represented a better path forward on health care.

Democrats were more positive, though Murray told reporters that there were a few "irons" to work out, suggesting the deal may not be completely final. Senate Minority Leader Charles Schumer (D-N.Y.) said he was "pleased" with the deal and urged Republican leaders to take it up as soon as possible. He said the measure includes "anti-sabotage" measures, an apparent reference to restoring the outreach funding.

Even with Trump's support, it's not entirely clear that a deal negotiated by Alexander and Murray can get through Congress.

When Republicans in the Senate made a last-ditch effort to repeal ObamaCare in September, the Alexander-Murray talks were cut off, with some GOP senators saying their bipartisan efforts would go nowhere in the House.

The chairman of the conservative Republican Study Committee caucus quickly criticized the deal, arguing it would "prop up" the existing health-care law.

"Anything propping it up is only saving what Republicans promised to dismantle," said Rep. Mark Walker (R-N.C.), the caucus chairman.