WASHINGTON — The White House’s aggressive push for Congressional approval of an attack on Syria appeared to have won the tentative support of one of President Obama’s most hawkish critics, Senator John McCain, who said Monday that he would back a limited strike if the president did more to arm the Syrian rebels and the attack was punishing enough to weaken the Syrian military.

In an hourlong meeting at the White House, said Mr. McCain, Republican of Arizona, Mr. Obama gave general support to doing more for the Syrian rebels, although no specifics were agreed upon. Officials said that in the same conversation, which included Senator Lindsey Graham, the South Carolina Republican, Mr. Obama indicated that a covert effort by the United States to arm and train Syrian rebels was beginning to yield results: the first 50-man cell of fighters, who have been trained by the C.I.A., was beginning to sneak into Syria.

There appeared to be broad agreement with the president, Mr. McCain and Mr. Graham said, that any attack on Syria should be to “degrade” the Syrian government’s delivery systems. Such a strike could include aircraft, artillery and the kind of rockets that the Obama administration says the forces of President Bashar al-Assad used to carry out an Aug. 21 sarin attack in the Damascus suburbs that killed more than 1,400 people.

The senators said they planned to meet with Susan E. Rice, Mr. Obama’s national security adviser, to discuss the strategy in greater depth.