While Trump called the legislation "popular," opinion polls have broadly shown that more respondents disapprove of the GOP plan than approve of it. Republicans contend that more people will warm to the tax bill when they see its effects next year.
Legislationunveiled by the House on Thursday would fund the government through Jan. 19. If Congress cannot pass a funding measure, the government will shut down early Saturday.
Ryan said Congress will push to pass a "clean" short-term continuing resolution funding bill to avert the shutdown.
"We're just bringing a clean, what we call vanilla CR — no games, no sneaky things. Just a continuing resolution to get us through this moment to get us into next year," Ryan told CNBC. "It's as clean and simple as possible. It's what the Senate on both sides of the aisle are asking us for."
The legislation Congress hopes to pass this week includes $2.85 billion in funding for the Children's Health Insurance Program. It would also include money for diabetes programs and community health centers, as well as extend the foreign surveillance warrant program.
Senate Majority Leader Mitch McConnell on Thursday said he does not think members of Congress want to see a shutdown.
"No one wants to do it," he said at an event hosted by news site Axios.
Democrats have pushed to protect spending on social safety net and health-care programs as part of a bill to fund the government. They have also pushed for protections for hundreds of thousands of young immigrants brought to the United States illegally as children.
"Republicans have spent the entire year trying to take health care away from millions and raise taxes on the middle class," said Drew Hammill, a spokesman for House Minority Leader Nancy Pelosi. "Democrats are simply asking for action on the broad set of bipartisan priorities before the Congress."
He added: "Republicans hold the House, the Senate and the White House. If they need Democratic votes, then they should incorporate the bipartisan proposals we are calling for into the continuing resolution."
A spokesperson for the Senate's top Democrat, Minority Leader Chuck Schumer, did not immediately respond to requests to comment on Ryan's remarks.