In interview this past weekend, McConnell gave Hagel a poor grade for his testimony before the Senate Armed Services Committee several days ago.
The Senate Armed Services Committee could vote as early as Thursday on the nomination and refer it to the full Senate. Democrats hold a 14-12 edge on the panel.
McConnell said opposition to the former Republican senator leading the Pentagon was growing.
"Whether that means he will end up having to achieve 60 votes or 51 is not clear yet," the Kentucky senator, who is seeking re-election, said Saturday at the opening of his campaign headquarters in Louisville.
Hagel seemed ill-prepared under withering cross-examination from Senate Armed Services Committee Republicans in nearly eight hours of testimony at his confirmation hearing last Thursday. He was repeatedly pressed about past statements and votes on Israel, Iran and nuclear weapons, with GOP lawmakers suggesting he wasn't sufficiently supportive of Israel or anti-Iran.
Senate Democrats, who hold the majority in the Senate, continue to stand behind the nomination, and no Democrat has said he or she would vote against the president's pick for his second-term national security team. Hagel, 66, is a decorated Vietnam combat veteran who served two terms as Nebraska senator.
About a dozen Republican have said they will oppose their former colleague and several others have indicated they are likely to vote no.
Democrats hold a 55-45 advantage in the Senate and two Republicans have announced their support for Hagel— Sens. Thad Cochran of Mississippi and Mike Johanns of Hagel's home state of Nebraska. Sen. Roy Blunt, R-Mo., a member of the leadership, said he would oppose Hagel, but would not back a filibuster.
McConnell's failure to rule out a filibuster marks the first time the leader has suggested it could be an option. In the past, when Republicans occupied the White House, GOP senators have argued strenuously that nominees should get an up-or-down Senate vote, especially Cabinet picks.
Last Friday, White House spokesman Jay Carney said the administration would be "stunned if, in the end, Republican senators choose to try to block the nomination of a decorated war veteran who was once among their colleagues in the Senate as a Republican."
The White House insisted on Monday that the president stands behind the nominee.
"The issue here is 'how will an individual do the job,' and there's no question in the president's mind that Sen.Hagel will do the job well," Carney said aboard Air Force One.