White House official says shutdown wouldn't stop ObamaCare enrollment

That means museums on the National Mall would close, passport services would be suspended, and trash collection in the District of Columbia would be halted.

Oct. 1 is also the first day the new federal health exchanges mandated under the Affordable Care Act will begin signing up new members. But, Simas said, registrations would proceed despite a potential shutdown because the large majority of Affordable Care Act spending would not be affected by a shutdown.

"There may be some impacts ... but it does not go into the core of what we are doing," Simas said.

The White House is in the midst of a communications blitz intended to raise public awareness about the exchanges. On Tuesday, President Obama and former President Clinton touted advantages of the law at an event in New York City. On Thursday, Obama will travel to Maryland for an event promoting the insurance exchanges.

Earlier Tuesday, the Department of Health and Human Services released a report showing that premiums on the exchanges are lower than had been projected. And President Obama will hold a conference call with state and local officials Tuesday afternoon to ask for help in promoting the sign-up.