WASHINGTON — Anxious congressional Democrats are threatening to abandon President Obama on a central element of his signature health care law, voicing increasing support for proposals that would allow Americans who are losing their health insurance coverage because of the Affordable Care Act to retain it.

The dissent comes as the Obama administration released enrollment figures on Wednesday that fell far short of expectations, and as House Republicans continued their sharp criticism of administration officials at congressional hearings examining the performance of the health care website and possible security risks of the online insurance exchanges.

In addition, a vote is scheduled Friday in the Republican-controlled House on a bill that would allow Americans to keep their existing health coverage through 2014 without penalties. The measure, drafted by Representative Fred Upton, the Michigan Republican who is the chairman of the Energy and Commerce Committee, is opposed by the White House, which argues that it would severely undermine the Affordable Care Act by allowing insurance companies to continue to sell health coverage that does not meet the higher standard of Mr. Obama’s health care law.

But a growing number of House Democrats, reflecting a strong political backlash to the rollout of the law, are warning the White House that they may support the measure if the administration does not provide a strong alternative argument. The approaching House vote is shaping up as an important test for both the health measure and the unity that Democratic leaders have so far been able to maintain around it despite a fierce Republican attack.