WASHINGTON - Senate Democrats said they neared agreement Saturday on a major overhaul of the nation’s health care system, putting them within reach of approving legislation by Christmas.

As the Senate convened in a driving snowstorm, Democratic lawmakers and senior officials said a breakthrough came when Senator Ben Nelson, Democrat of Nebraska, agreed after hours of negotiation Friday to back the legislation, making him the pivotal 60th vote.

In a sign of confidence that he had the necessary support, Senator Harry Reid of Nevada, the majority leader, introduced his 383-page proposal in hopes of overcoming a series of procedural hurdles over the next several days. Republicans immediately forced a reading of the measure, which was expected to take at least 10 hours.

Mr. Reid’s amendment includes major restrictions on abortion that were intended to win support for the bill from Mr. Nelson. Under Mr. Reid’s proposal, health insurance plans are not required or forbidden to cover abortion services, but there is a major exemption that would give states power to prohibit abortion coverage in the insurance markets, or exchanges, where most health plans would be sold.

Mr. Reid’s amendment also includes a substantial increase in federal contributions to Nebraska’s costs of providing Medicaid coverage to the poor.

Republicans remained deeply opposed to the proposal and Senator Olympia J. Snowe, a Maine Republican who had been considered a possible Democratic ally, said she would oppose the measure, saying it was being rushed through.

“It is a take-it-or leave it package,” she said.