WASHINGTON — The Obama administration on Tuesday provided the first detailed look at premiums to be charged to consumers for health insurance in 36 states where the federal government will run new insurance markets starting next week, highlighting costs it said were generally lower than previous estimates.

Administration officials released the information, central to their campaign to persuade millions of uninsured Americans to sign up for coverage, even as Senator Ted Cruz, Republican of Texas, waged a fierce fight on the Senate floor, risking a government shutdown if necessary to eliminate financing for the expansion of coverage under President Obama’s health care law.

The White House sought to focus attention on what it portrayed as the financial advantages of the health insurance program, which is set to start accepting customers on Oct. 1.

“I can tell you right now that in many states across the country, if you’re, say, a 27-year-old young woman, don’t have health insurance, you get on that exchange, you’re going to be able to purchase high-quality health insurance for less than the cost of your cellphone bill,” Mr. Obama said Tuesday, speaking at a health care forum in New York City with former President Bill Clinton.