The messages — “paid for by the U.S. Department of Health and Human Services” — urge beneficiaries to “check out Medicare Advantage” and point to an online tool, the Medicare plan finder, to compare the different options.

“You may be able to lower your out-of-pocket costs while getting extra benefits, like vision, hearing, dental and prescription coverage,” said an email sent to beneficiaries on Wednesday.

“With Medicare Advantage,” says another email, “one plan covers all of your care.”

In small print, the emails say they were “created and distributed by the Centers for Medicare and Medicaid Services” to people who “signed up for email updates from the Medicare team.”

Seema Verma, the administrator of the Centers for Medicare and Medicaid Services, said the agency was not favoring private plans over the original government-run Medicare program.

“We are not steering any Medicare beneficiary anywhere,” she said.

But Richard S. Foster, who was for many years the nonpartisan chief actuary of the Medicare program, said the emails sounded “more like Medicare Advantage plan advertising than objective information from a public agency.”