Apart from voters, the GOP could also face pressure from the White House to act.
Trump "remains committed to providing the American people with affordable healthcare, including relief from the onerous mandates and taxes of the catastrophic Obamacare law," Deputy White House Press Secretary Hogan Gidley said in a statement.
Republican senators say they hope to advance two bipartisan proposals this year to help bring some relief to Americans in advance of the 2018 elections.
One from Senate Health, Education, Labor and Pensions Chairman Lamar Alexander and ranking member Patty Murray would give states greater flexibility to implement aspects of the law.
But while that measure enjoyed strong bipartisan support last year, it could face more difficult prospects this year with Democrats now demanding changes to the legislation.
"There's not a way for the language that we agreed on to deal with the marketplace that has changed," Murray said.
Senators are hoping to attach the Alexander-Murray measure, as well as a separate bill from Maine Republican Susan Collins and Florida Democrat Bill Nelson, to a fiscal 2018 spending bill.
"There was a commitment to do that by the president. Sen. McConnell's made that commitment, so that's our hope," Senate Majority Whip John Cornyn of Texas said.
But discussions on health care continue to be overshadowed by several other issues, including the debate over the pending expiration of the Deferred Action for Childhood Arrivals program, funding the government past Jan. 19, and addressing the debt ceiling.
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