WASHINGTON — With lawmakers poised to begin a major push to reshape the nation’s health-care system, President Barack Obama on Wednesday signaled new openness to the idea of the government requiring most Americans obtain medical insurance — a position likely to increase momentum behind the drive to create a coverage mandate.

At the same time, the president, who rejected such a mandate during the 2008 campaign, reaffirmed in strong terms his determination to offer a government-run health-care plan as an alternative to private insurance.

“I strongly believe that Americans should have the choice of a public health insurance option operating alongside private plans,” Obama said in a two-page letter to Sens. Edward Kennedy, D-Mass., and Max Baucus, D-Mont., who are leading efforts to develop health-care legislation. “This will give them a better range of choices, make the health-care market more competitive and keep insurance companies honest.”

Obama also indicated he was receptive to a new requirement that large businesses share in the cost of providing health insurance. And he called for at least another $200 billion in cuts to the federal Medicare and Medicaid insurance programs for senior citizens and poor families to pay for expanded coverage. This would be in addition to $300 billion in cuts requested earlier.

The letter, released a day after the president met with two dozen Senate Democrats at the White House to talk about health care, came as the White House intensified its effort to build momentum for a major health-care overhaul this year. Obama has asked Congress to send him a bill by October.

Kennedy and Baucus plan to introduce major health-care bills over the next several weeks, as do senior House Democrats. At the same time, many Republicans are stepping up their criticism of the initiative, which they claim will increase government control of medicine.

Obama and his chief lieutenants have emphasized the need to control costs in reaction to widespread public anxiety about rising health-care bills — as well as concerns among experts about the nation’s skyrocketing health-care tab, which this year will top $2.2 trillion.

“I want to stress that reform cannot mean focusing on expanded coverage alone,” Obama wrote. “Indeed, without a serious, sustained effort to reduce the growth rate of health-care costs, affordable health-care coverage will remain out of reach.”

Though Obama has made a health-care overhaul one of his top priorities, he has thus far largely avoided staking out firm positions on some of the most contentious issues, endorsing broader goals while letting senior Democrats write the specifics on Capitol Hill.

The coverage mandates, described sometimes as “shared responsibility,” and the so-called public plan have been among the more controversial concepts in the emerging debate.

In his letter, the president left room for compromise on the mandates, suggesting exemptions for people who cannot afford insurance and for small businesses.

At least one GOP lawmaker praised the president’s approach.

“Having the president engaged in the legislative debate with yesterday’s meeting and today’s letter, which doesn’t draw lines in the sand, is helpful because of his statements that reform legislation needs to have bipartisan support,” said Iowa Sen. Charles Grassley, the senior Republican on the Senate Finance Committee.

But Obama’s letter, which came as lawmakers are finalizing their health-care bills, represented a direct challenge to business groups long wary of requirements that they provide insurance or pay some kind of fee to help the government finance wider coverage.

A spokeswoman for the U.S. Chamber of Commerce called the mandate a “job killer.” Even some small businesses are wary of mandates with exemptions, out of concern that the exemptions could be adjusted in the future.

Insurance companies, which have embraced a new mandate, vigorously oppose the proposed government insurance program.

The president also waded further into the potentially explosive debate over how to pay for an overhaul that is expected to cost more than $1 trillion over the next decade.

Obama pledged to work with Congress to cut $200 billion to $300 billion from the Medicare and Medicaid insurance programs over the next 10 years.

That would come on top of $634 billion the president proposed in his budget to pay for health reform.

Roughly half of that came from changes in Medicare, including cuts to private insurers who contract with the government to provide insurance to seniors. The rest would come from reducing the amount wealthiest Americans could deduct on their taxes for charitable donations, state taxes or interest payments on a home.

That proposal has run into stiff opposition from Democrats as well as Republicans, but administration officials have not given up on it. And Obama signaled his intent to keep it on the table as legislative debate proceeds.

The president also suggested empowering a federal commission under the Department of Health and Human Services to make further cuts in Medicare spending with less interference from Congress.