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Despite the deadline, Democrats were content to let Mr. Cruz and his allies delay.

If they use all of the time possible, it would mean the Senate would hold a final vote late Sunday, throwing the bill back to the House with little more than 24 hours before the Monday midnight deadline.

The health care law is growing more unpopular, according to opinion polls that suggest the ongoing fight is taking a toll on Mr. Obama’s signature achievement.

Still, polls show voters don’t want a government shutdown and say the spending fight should be kept separate.

A United Technologies/National Journal Congressional Connection poll released this week found 63 percent of Americans oppose holding up government spending until Obamacare is defunded.

Republicans counter that Congress‘ best leverage is with the spending bills — or with the debt limit battle, which looms next month.

While the fight rages in Washington, states across the country are preparing to open their health care exchanges, or marketplaces, where those without insurance can buy plans, often with the help of government subsidies.

Although the bulk of the law has been held to be constitutional by the Supreme Court, businesses have challenged an Obama administration directive requiring them to provide contraceptive coverage, including sterilization and the morning-after pill, in plans they offer their employees. The businesses argue that the provision violates their conscience rights.

Appeals courts have split on the issue, making it likely that the Supreme Court ultimately will decide the case.

Jacqueline Klimas and Seth McLaughlin contributed to this report.