Federal officials are planning a workaround that would effectively extend the March 31 deadline to enroll for health-care coverage for some users if technical glitches hamper a last-minute surge of signups on HealthCare.gov, people familiar with the matter say.

Despite improvements since its rocky Oct. 1 launch, the website still faces problems, including some involving the same software that stymied users in the fall. Problems with the site at least three times this past week kept some users from being able to log on, including...