McCain, who has been at the center of another heated Washington debate over the confirmation of President Barack Obama's nominee for defense secretary, acknowledged Republicans struggled with Hispanic voters in November's election, asked whether Democrats were serious about passing comprehensive immigration reform -- and taking a winning issue off the table in 2014's midterm election.
"I believe we're making progress on a bipartisan basis," McCain said on NBC's "Meet the Press" Sunday.
McCain is one of a group of eight bipartisan senators working to come up with an immigration bill that appeals to members of both parties.
"Does the president really want a result?" McCain said. "Or does he want another cudgel so he can beat up Republicans to get an advantage in the next election?"
McCain and other GOP senators voted to postpone Hagel's confirmation last week, saying they needed more time to pore over documents related to Obama's response to a deadly attack on the U.S. consulate in Benghazi, Libya -- though Hagel himself was acting as a college professor at the time of the incident and had no official role in the response.
McCain acknowledged Hagel likely has the votes to be confirmed when the Senate returns to business next week.
Obama Chief of Staff Denis McDonough said Hagel will be ready to serve if and when he is confirmed.
"No, he's not going to be a weaker defense secretary," McDonough said. "He's going to be a great defense secretary."