WASHINGTON — Millions of seniors enrolled in some of the most popular Medicare prescription-drug plans face double-digit premium hikes next year if they don’t shop for a better deal, said a private firm that analyzes the highly competitive market.

Seven of the top 10 prescription plans are raising their premiums by 11 percent to 23 percent, according to a report this week by Avalere Health.

It’s a reality check on a stream of upbeat Medicare announcements from the Obama administration, all against the backdrop of a hard-fought election. In August, officials had announced that the average premium for basic prescription-drug coverage will stay the same in 2013, at $30 a month.

The administration’s number is accurate as an overall indicator for the entire market, but not very helpful to consumers individually since it doesn’t reflect price swings in the real world.

“The average senior is going to benefit by carefully scrutinizing their situation, because every year the market changes,” Avalere President Dan Mendelson said. Avalere crunched the numbers based on bid documents that the plans submitted to Medicare.

The report found premium increases for all top 10 prescription-drug plans, known as PDPs. However, the most popular plan — AARP MedicareRx Preferred — only is going up 57 cents per month nationally, to $40.42 from the current $39.85.

President Barack Obama’s health-care law does not appear to be the cause of the increases. Indeed, the law is improving the prescription benefit by gradually closing a coverage gap called the “doughnut hole,” which catches people with high drug costs.

Instead, the price hikes appear to be driven by market dynamics, and some insurers are introducing new low-premium options to gain a competitive advantage on plans that are raising their prices.

The seven plans with double-digit premium increases were: the Humana Walmart-Preferred Rx Plan (23 percent); First Health Part D Premier (18 percent); First Health Part D Value Plus (17 percent); Cigna Medicare Rx Plan One (15 percent); Express Scripts Medicare-Value (13 percent); the HealthSpring Prescription Drug Plan (12 percent); and Humana Enhanced (11 percent).

Another two plans in the top 10 also had single-digit increases. They were the SilverScript Basic (8 percent) and WellCare Classic (3 percent).

On the plus side for consumers, a major new low-cost plan entered the market.

Premiums for the AARP MedicareRx Saver Plus Plan will average $15 a month nationally, although it won’t be available everywhere. That’s $3.50 less than the current low-cost leader, the Humana Walmart plan, whose premiums are rising to $18.50.