Sen. Lindsey Graham said Monday that Democrats will change their tune about repealing and replacing Obamacare “when they lose elections.”

Mr. Graham, South Carolina Republican, made his prediction to Fox News as the Obama administration scrambles to get as many enrollees as possible to pick a private health plan before the overhaul’s deadline late Monday.

While the White House sounded upbeat about the law’s prospects, GOP operatives and lawmakers doubled down on their belief that the law’s troubles will sway November’s midterm elections.

“The worst is yet to come,” Mr. Graham said, saying Democrats’ shaky promises about the law amount to “a trust issue.”

He said Republicans should retain some elements of the health care law, such as letting young adults stay on their parents’ plans until age 26, but dismantle most of the law and start over with a workable plan.