WASHINGTON  President Obama, who has stood by his landmark health care law through court attacks and legislative efforts to repeal it, told the nation’s governors on Monday that he was willing to amend the measure to give states the ability to opt out of its most controversial requirements right from the start, including the mandate that most people buy insurance.

In remarks to the National Governors Association, Mr. Obama said he supported legislation that would allow states to obtain waivers from the mandate as soon as it took effect in 2014, as long as they could find another way to expand coverage without driving up health care costs. Under the current law, states must wait until 2017 to obtain waivers.

The announcement is the first time Mr. Obama has called for altering a central component of his signature health care law, although he has backed removing a specific tax provision that both parties regard as onerous on business.

But the prospects for the proposal appear dim. Congress would have to approve the change through legislation, and House Republican leaders said Monday that they were committed to repealing the law, not amending it. Even if the change were approved, it could be difficult for states to meet the federal requirements for the waivers.