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Legal scholars have questioned whether Mr. Obama has that sort of broad authority on deportations, and Thursday’s move reignited that debate.

“It is true that the chief executive has some room to decide how strongly to enforce a law, and the timing of enforcement. But here, Obama is apparently suspending the enforcement of a law for a year — simply to head off actual legislation not to his liking,” Eugene Kontorovich, a professor at Northwestern University School of Law, said in a blog post at Volokh Conspiracy.

“The ‘fix’ amounts to new legislation — but enacted without Congress,” Mr. Kontorovich said. “The president has no constitutional authority to rewrite statutes, especially in ways that impose new obligations on people, and that is what the fix seems to entail.”

Mr. Obama’s Democratic allies on Capitol Hill, however, said they believe he acted within his power and has done enough to solve the problem.

“There is no need for a legislative fix for this issue,” said Senate Majority Whip Richard J. Durbin, Illinois Democrat.

Republicans, though, counter that even if it’s legal, Mr. Obama’s fix is temporary and he could change it at any time — meaning insurance companies lack the kind of certainty they need to do business.