The New York Democrat, who has been non-committal in his public statements about Hagel's nomination, is expected to meet with Hagel this week and press the former Republican senator from Nebraska on issues concerning Iran and Israel, the Times said.
Schumer has told aides and other senators he could be persuaded to support Hagel if he is satisfied with the meeting's outcome, the Times said.
Not getting the support of Schumer -- widely seen as the Senate's most powerful Jewish Democrat -- would likely end Hagel's candidacy, Politico said.
Schumer's views would likely influence other Democrats, Politico and the Times said, and give political cover to Republicans fighting Hagel's nomination.
Sen. John McCain of Arizona -- the ranking Republican on the Armed Services Committee, which will conduct Hagel's confirmation hearing -- told "Face the Nation" Sunday Hagel's early opposition to the troop surge in the Iraq War was "bizarre."
Hagel and McCain both fought in the Vietnam War. Hagel received the Purple Heart twice for wounds while an infantry squad leader. McCain was also wounded and was captured by the North Vietnamese, who held him as a prisoner of war for six years.
Schumer plans to ask Hagel to clarify and perhaps recant statements about Iran and Israel, a person with knowledge of Schumer's plans told the Times.
Hagel has said a military strike against Iran to stop its nuclear threat would be counterproductive. He also made statements indicating he would accept Iran's nuclear-weapons development while seeking to prevent a Middle East nuclear arms race -- statements that contradict Obama and Schumer positions.
Hagel was among a few senators who wouldn't sign a letter to the European Union calling for the designation of Hezbollah as a terrorist organization.
Hagel has since said he didn't sign it because he does not believe in sending letters to foreign governments. He has also supported direct talks between the United States and Hamas, which governs Gaza.
The United States, Israel, Canada, the European Union and Japan classify Hamas as a terrorist group. The United States, Israel, Canada, Britain, the Netherlands, Bahrain, Egypt and Australia classify Hezbollah as a terrorist organization, in whole or in part.
Schumer is also expected to ask Hagel about a 1998 comment when he questioned whether "openly, aggressively gay" James Hormel was appropriate to be President Bill Clinton's ambassador to Luxembourg.
Hagel apologized for that comment last month, saying he also supported gays in the military -- a policy he once opposed.
Schumer was also expected to ask Hagel about his views on abortion, particularly his opposition to allowing abortions for military women who have been raped, the Times said.
Powell told the NBC News program "Meet the Press" he considered Hagel "superbly qualified" to replace retiring Defense Secretary Leon Panetta.
"First, I think he's had a very, very distinguished public service record that he can stand on," Powell said. "There are a lot of comments about different things he said over the years and I think he will have a chance to respond to all those comments as the confirmation hearings."
But Sen. Bob Corker, R-Tenn., told the ABC News program "This Week" he had questions about Hagel's "overall temperament, and is he suited to run a department or a big agency or a big entity like the Pentagon."
Corker, who is not on the Armed Services Committee and said he did not know Hagel well, said: "I think there are numbers of staffers who are coming forth now just talking about the way he has dealt with them. I have certainly questions about a lot of things."