"With the Supreme Court pushing to the front of the agenda in the Senate and jobs, terror and debt being our major issues we should be worrying about, this is a treaty for next year," Alexander said on "Fox News Sunday."
"There's not a chance the treaty will be approved this year," said Alexander, the third-highest ranking Republican in the Senate.
The Obama administration said Sunday a deal with Russia to cut nuclear arsenals and new limits on the use of U.S. nuclear weapons would not pose security risks.
President Barack Obama signed the arms-control treaty with Russian President Dmitry Medvedev Tuesday. The same day, Obama unveiled a new nuclear strategy that says the United States would not use nuclear weapons against any non-nuclear country that is in compliance with non-proliferation agreements.
The administration's Nuclear Posture Review labels nuclear terrorism the greatest threat to U.S. and global security, and Obama said security for the United States and its allies can be increasingly defended "by America's unsurpassed conventional military capabilities and strong missile defenses."
Treaty ratification requires 67 votes in the Senate, meaning Democrats would need the support of at least eight Republicans.
Also appearing on Fox, Connecticut Independent Sen. Joseph Lieberman said Obama would have trouble winning Senate approval of the treaty unless the administration pledged to modernize the U.S. nuclear weapons stockpile.
"As we reduce the number of nuclear warheads America has in a world that is still dangerous, very dangerous, and in which the threat of the spread of nuclear powers, particularly Iran, grows every day, we have to make darn sure that our nuclear warheads are capable, are modern," Lieberman said.
"I'm going to be real hesitant to vote for this treaty unless we have a commitment from the administration that they're prepared to modernize our nuclear stockpile," he said.