That potential record for enrollment levels comes despite opposition to Obamacare by the Trump administration, and despite sharp defunding of sign-up outreach efforts and advertising by the administration.
TheHill.com reported that last Wednesday, the first day of open enrollment for individual health plans, more than 200,000 people selected an Obamacare plan for 2018.
That compares with only around 100,000 for the same day last year, according to TheHill.com, citing a source close to the enrollment process.
There was also a sharp uptick in visits to HealthCare.gov, the federally run insurance marketplace that sells Obamacare plans in much of the United States.
More than 1 million people visited that sign-up site last Wednesday, compared with about 750,000 visitors on the first day of open enrollment in 2016.
Earlier this week, ACASignups.net, which tracks Obamacare-related data, reported that enrollment activity on Covered California, the largest state-run health exchange in the nation, was also markedly higher.
Last Wednesday, nearly 6,000 people selected a plan on Covered California, about 25 percent more than did on the same day last year, ACASignups.net said.
Enrollments on another state-run exchange, Maryland Health Connection, were up even more — by 70 percent — on opening day.
The Baltimore Sun reported that exchange officials said that more than 1,800 people signed up on Maryland's marketplace, compared with 1,055 on the first day of enrollment last year.
HealthSherpa.com, an online health insurance brokerage, said it had seen a 200 percent increase in the number of enrollments submitted on its site last Wednesday compared with the first day of open enrollment in 2016.
George Kalogeropoulos, CEO of HealthSherpa.com, said that a major reason for that increase in sign-ups was recently approved technology that made it easier and faster for web-based brokers to enroll Obamacare customers in plans sold on HealthCare.gov.
A spokeswoman for the U.S. Department of Health and Human Services, which oversees Obamacare, declined to comment to CNBC when asked for an official sign-up tally for the past week, and how it compares with the same time period last year.
The spokeswoman noted that the Centers for Medicare and Medicaid Services, an HHS division, plans to release snapshot enrollment data throughout the sign-up season as CMS has done in the past.
Lori Lodes, who had been communications director at CMS during the Obama administration, told CNBC, "The great start to open enrollment proves just how much people value health coverage and that people are finding coverage they can afford."
"There's only five weeks to go before the final deadline of Dec. 15 so we've got a lot of work to do to enroll even more people," said Lodes, referring to the sign-up deadline for HealthCare.gov.
Other states, including California and New York, have deadlines weeks later.
"Imagine how many more people would be signing up if the administration stopped their efforts to undermine the law and actually got the word out about open enrollment," said Lodes, who co-founded a group called Get America Covered to promote sign-ups this season.
The Trump administration in late August said it would slash the Obamacare advertising budget by 90 percent, and the budget for in-person enrollment by 41 percent.
In September, the administration told HHS' 10 regional directors not to participate in state-based events promoting Obamacare enrollment, after years of having done so.
Those moves still could lead to lower enrollment on Obamacare exchanges by the end of the sign-up season. The exchanges sell health coverage, often at subsidized discounts, to people who do not have insurance through a job, Medicare, Medicaid, or other sources.
Lodes' fellow co-founder at Get American Covered, Joshua Peck, recently estimated that the administration's pullback on enrollment promotion efforts will lead to at least 1.1 million fewer people signing up for Obamacare plans this season compared with last season.
That estimate by Peck, former chief marketing officer of HealthCare.gov, is a best-case scenario.
Peck told CNBC that the impact of the Trump administration's moves "is pretty grim."