Any nominee must have "a full understanding of our close relationship with our Israeli allies, the Iranian threat and the importance of having a robust military," McConnell said on ABC's "This Week."
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."
Some congressional Democrats also share the GOP concerns, including Maryland Sen. Ben Cardin, who called the pick controversial and said Hagel needs to clarify his comments on Iran and Israel.
"There are Democrats including this senator who have questions that have to be answered before I can support him," Cardin said in an interview with Current TV's "Bill Press Show." He added that the Senate will probably confirm Hagel.
While Hagel will face opposition, Democrats privately say they do not foresee a long, protracted fight similar to what President George H.W. Bush faced in 1989 when he selected former Sen. John Tower to serve as defense secretary. Democrats led the charge in scuttling that nomination, a rare defeat for a new president. Questions about Tower's drinking and womanizing sank his nomination.
Monday's nominations leave Obama without a woman in line for a top administration post, a fact that has irked some Democratic women. The president will soon name a new treasury secretary, but current White House chief of staff Jack Lew is the front-runner for the post.
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Associated Press writers Robert Burns, Donna Cassata and Matthew Daly contributed to this report.