The two reports suggest that enrollment through the federal and state exchanges is sharply lagging the administration's target of 7 million people by the end of March.
The federal and state exchanges are intended to be the gateway for private health plan coverage for some 30 million uninsured Americans under the health-reform law known as the Affordable Care Act.
Administration officials are expected to release official enrollment data sometime this week.
Meanwhile, some insurance companies are pushing the administration for an end-run around the hobbled HealthCare.gov exchange to allow them to directly enroll people entitled to federal subsidies, The New York Times reported.
The Times said the administration has so far resisted the idea, because of concerns about giving insurance companies access to personal data. But, the newspaper added, senior White House officials may now be more open to ways to allowing insurers to handle more enrollments because of the federal website's technical problems.
Chris Jennings, a senior health care adviser to President Obama, said in a statement that the administration was "continuing to pursue additional avenues by which people can enroll, such as direct enrollment through insurance companies, that will help meet pent-up demand," the Times reported.
State websites, overall, have experienced fewer problems than the federal health insurance exchange. Yet compared with early enrollment in Medicare's prescription drug program, which took effect in 2006, state-based enrollment through the Affordable Care Act is lagging.
"When Medicare Part D began, two-thirds of people who voluntarily enrolled in a plan did so after coverage began on Jan. 1," Avalere reported.