Roughly 40,000 Americans have signed up for private insurance through the flawed federal online insurance marketplace since it opened six weeks ago, according to two people with access to the figures.

That amount is a tiny fraction of the total projected enrollment for the 36 states where the federal government is running the online health care exchange, indicating the slow start to the president's initiative.

The first concrete evidence of the popularity — and accessibility — of the new federal insurance exchange emerged as the White House has been preparing to release this week the first official tally of how many people have chosen coverage using the website, HealthCare.gov.