US President Barack Obama pauses while speaking at the White House Youth Summit on the Affordable Care Act in Washington, in this Dec 4, 2013 file photo. [Photo/Agencies]

WASHINGTON - The US government said it was still processing thousands of sign-ups for health insurance under President Barack Obama's signature healthcare law on Tuesday as Americans made a final rush to obtain medical coverage in time for New Year's Day.

Citing nearly 2 million visits to the HealthCare.gov website and over 250,000 inquiries at call centers before Monday's sign-up deadline, the government gave consumers an extra day to enroll by midnight on Tuesday for January coverage.

It added flexibility by encouraging consumers to contact government call centers if they had started but not been able to finish their applications, without specifying a deadline for completing those enrollments.

"Sometimes despite your best efforts, you might have run into delays caused by heavy traffic to HealthCare.gov, maintenance periods, or other issues with our systems that prevented you from finishing the process on time," read a blog post on HealthCare.gov on Tuesday. "If this happened to you, don't worry -- we still may be able to help you get covered as soon as January 1."

The last-minute changes were aimed at getting as many people as possible insured under the Affordable Care Act, Obama's major domestic policy initiative. The administration has been moving the sign-up goal posts as it tries to recover from technical failures and political missteps that dogged the enrollment drive for weeks after it opened on October 1.

Ahead of Monday, more than 1 million people had signed up for private coverage through HealthCare.gov -- which serves 36 states -- and 14 state-run marketplaces, according to state and federal estimates.

Enrollment data from the final dash was not available on Tuesday, but administration officials cited "amazing interest in signing up for Jan 1."

"We are taking thousands of calls at our call centers, which remain open until midnight, and we are seeing thousands of visitors complete enrollment online," said Julie Bataille, spokeswoman for the Centers for Medicare and Medicaid Services (CMS), which runs the HealthCare.gov site.

Americans who wait until after Tuesday to begin the process of selecting a health plan will have missed the deadline for health insurance coverage starting January 1 and will be covered as of February 1, CMS said on Tuesday afternoon.

The New Year's push represents the biggest test yet of the program's ability to draw in enough consumers to make it financially viable. The Congressional Budget Office forecast 7 million people would sign up by March 31, the last date to get coverage in 2014, but HealthCare.gov's problems threw those estimates into doubt.

About one-third or more of those newly insured will need to be young, healthy adults whose payments into the system would help offset the costs of covering older, sicker people.