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“That is a 250,000-vote net increase for a state Obama won by 260,000 votes in 2008,” Mr. Jennings said.

And Republicans say their early voters include a higher percentage of people who don’t often vote. They argued that more of the Democrats who are voting early in Ohio are people who would have gone to the polls on Election Day regardless. Former George W. Bush adviser Karl Rove said on “Fox News Sunday” that Republicans are turning out more “new” voters in Ohio while Democrats are essentially “cannibalizing” their Election Day voters.

Mr. Messina said the Obama campaign is “gaining steam in Ohio.”

“There have been 16 polls out; we’ve led in 14 and are tied in the other two,” he said.

The final week of the race saw both campaigns making moves in Pennsylvania, a state the candidates have largely avoided as Mr. Obama had a comfortable lead in most polls until early October. Since Mr. Romney closed the gap in Pennsylvania after the first debate Oct. 3, however, the Support Our Future PAC has decided to spend $2.1 million on ads there.

In response, the Obama campaign said it would pay for its own ad buy in Pennsylvania. Mr. Messina said the move by Mr. Romney’s supporters is an indication that Republicans know they can’t win Ohio, and they are “desperate” to win another state.

“We are going to win Pennsylvania,” Mr. Messina said. “But we’re not going to take anything for granted.”

The Republican Party in Pennsylvania said Monday that it is leading in absentee ballots cast by 18.8 percentage points, 55.2 percent to 36.4 percent. Four years ago, the Republicans led by only 1.9 percentage points in absentees.