Mrs. Clinton’s afternoon rally at a community college in Cleveland, the heart of Democratic strength in Ohio, was aimed at increasing early voting, which began last week.

While early voting has traditionally been a Democratic strength in Ohio, the data shows potential complications for Mrs. Clinton on this front. Requests for early ballots are down 22.3 percent in Cuyahoga County, where Cleveland is, compared with the same period in 2012. Requests are also off 12.7 percent in Franklin County, which includes Columbus, the capital, according to data compiled by Michael McDonald, a University of Florida professor who studies voter turnout.

A decided advantage for Mrs. Clinton here is the strength of Democrats’ grass-roots efforts. The party has leased 93 offices statewide, according to Chris Wyant, the Clinton campaign director in Ohio.

Mr. Trump’s Ohio director cut ties recently with the state’s Republican chairman, Matt Borges, over Mr. Borges’s sharply worded public doubts about Mr. Trump. “They have an internal civil war on the Republican side,” Brian Fallon, Mrs. Clinton’s press secretary, told reporters here.