Hagel has criticized discussion of a military strike by either the U.S. or Israel against Iran. He has also backed efforts to bring Iran to the table for talks on future peace in Afghanistan. Some lawmakers have been troubled by his comments and actions on Israel, including his reference to the "Jewish lobby" in the United States.
"This is a controversial pick," Sen. Lindsey Graham, R-S.C., told CNN. "He is an antagonistic figure when it comes to the state of Israel. It's a signal you're sending to Iran at the worst possible time and to our allies."
McConnell, R-Ky., said Hagel, who left the Senate in 2009, has "certainly been outspoken in foreign policy and defense over the years. The question we will be answering, if he's the nominee, is do his views make sense for that particular job?"
McConnell said he would "wait and see how the hearings go and see whether Chuck's views square with the job he would be nominated to do." He added, "I'm going to take a look at all the things that Chuck has said over the years and review that, and in terms of his qualifications to lead our nation's military."
The second-ranking Senate Republican, John Cornyn of Texas, said in a statement that making Hagel defense secretary would be "the worst possible message we could send to our friend Israel and the rest of our allies in the Middle East." Cornyn did not say he would try to block a Hagel nomination.
Obama, in an interview that aired last week on NBC's "Meet the Press," called Hagel "a patriot" who "has done extraordinary work" in the Senate and on an intelligence advisory board.
Illinois Sen. Dick Durbin, the second-ranking Senate Democrat, told CNN's "State of the Union" that Hagel "is a serious candidate if the president chooses to name him."
Associated Press writers Jack Gillum and Donna Cassata contributed to this report.