The alleged errors include charging enrollees too much for their insurance, steering them to the wrong health plan or denying them coverage entirely.
"We are going to get them help," White House spokesman Jay Carney told reporters.
He said 22,000 represented "a very small percentage" of the people who successfully applied for coverage through the federal marketplace and said most of the technical issues creating the errors had been corrected.
The Washington Post reported the issue Sunday, citing internal government data.
The newspaper said many of the 22,000 people trying through appeals to get their enrollment errors fixed were told the federal government's computer system didn't yet let workers go into enrollment records to change them.
The website, whose rollout started Oct. 1 despite a concurrent partial government shutdown, was initially marred by serious technological problems, making it hard for people to sign up for insurance.
"We believe that many of the issues that caused people to file appeals are left over from when the website was not working well, and many of those problems have since been fixed," Carney said.
The Centers for Medicare and Medicaid Services, the Department of Health and Human Services agency that oversees the federal online portal, "has been reaching out to people who have filed an appeal via phone and email, and so far have seen that many of the problems were related to those earlier system errors that have now been fixed," Carney said.
"In many cases they've been able to help these consumers move forward with a new application, and without having to continue their appeal," he said.
A CMS spokesman told the Post essentially the same thing, that it was "inviting those consumers back to HealthCare.gov, where they can reset and successfully finish their applications without needing to complete the appeals process."
But some people who already pay for their new insurance plans at a higher rate, due to HealthCare.gov miscalculations, would not be able to get the difference in their premiums back if they completed a new application, the Post said.
Carney said technical problems were being fixed and the website now functioned "as it should" for "the vast majority of people who use it."
That means people can search for and sign up for health plans, he said.
Some 3 million people were enrolled in private health insurance through the federal and state marketplaces as of Jan. 24, HHS said.