WASHINGTON — Senator John McCain, who is battling brain cancer, has returned home to Arizona and is likely to miss the Senate’s vote this week to approve a sweeping tax overhaul, though President Trump said on Sunday that the senator would return if his vote was needed.

Mr. McCain’s office said in a statement on Sunday night that the senator, who had been hospitalized recently in the Washington area, would undergo physical therapy and rehabilitation at the Mayo Clinic in Arizona and “looks forward to returning to Washington in January.”

“He is grateful for the excellent care he continues to receive, and appreciates the outpouring of support from people all over the country,” the statement said.

Mr. McCain disclosed in July that he had an aggressive form of brain cancer, for which the median survival is not much more than a year. His office had said on Wednesday that he was receiving treatment at Walter Reed National Military Medical Center for “normal side effects of his ongoing cancer therapy,” and he missed all of the Senate’s votes last week.