WASHINGTON—House Speaker Nancy Pelosi on Sunday expressed confidence she will have enough votes to pass a health overhaul, but acknowledged it could come at a political cost to lawmakers who back the measure.

Monday kicks off a critical week for Democrats to push ahead with their overhaul. President Barack Obama is expected to announce his preferred way forward for the bill. The White House has already laid the groundwork for Congress to complete the legislation using a process known as budget reconciliation that requires a simple majority in the Senate.That process would start in the House. Ms. Pelosi said Sunday the House could unveil specific legislative language for the measure in a matter of days. Those would give more detail to the president's proposed changes to the Senate bill designed to appeal to House lawmakers.

Democrats seem poised to move forward without any support from Republicans, who say they see no way to work from the existing bill. That leaves Democrats with two critical challenges: figuring out how to maneuver through the reconciliation over Republican objections, and how to rebuild sliding public support for the bill.