Donald Trump is projected to win the key state of Ohio, with 53.3 percent of the vote, CNN reports.
Other battleground states, including Florida, Virginia, North Carolina and Georgia, were still too close to call.
This would give Democrat candidate Hillary Clinton 109 of the 270 electoral votes needed to win, and Mr Trump would have 167, CNN reported.
In other recent states to be called, Mrs Clinton won New York and Connecticut. Mr Trump won Texas and Arkansas.
Early predicted victories
Earlier, Mrs Clinton got a series of expected wins, including in Vermont, Massachusetts, Maryland, Rhode Island, New Jersey, Illinois, Delaware and the District of Columbia.
South Carolina, Alabama, Indiana, West Virginia, Kentucky, Oklahoma, Tennessee and Mississippi, among others, went to her Republican rival.
The early victories for both in these states were long-predicted.
'I want to see everything honest' - Trump
Earlier, Mr Trump said he would have to "see how things play out" before accepting the final election results. He said he had seen reports of voting irregularities, including that machines were changing votes, but gave few details.
American voters reported long lines and isolated cases of harassment at polling places, but fears of bigger problems did not appear to have materialised.
Local media reported that touch-screen voting machines were not recording ballots correctly in several Pennsylvania counties.
Civil rights groups said voters had complained of intimidating behaviour at polling booths in Pennsylvania and Florida as supporters of both candidates cast their ballots.
A Democratic Party source said the Clinton campaign had not encountered systemic problems beyond the usual election day hiccups.
Mrs Clinton cast her ballot at an elementary school near her home in Chappaqua, New York early on Tuesday morning.
"It is the most humbling feeling. I know how much responsibility goes with this," Mrs Clinton said.
Mr Trump, who voted in Manhattan, said it was a tremendous honour to cast his ballot.
No presidential vote from Bush
Former Republican president George W Bush and his wife, Laura, did not cast a vote for US president on Tuesday but did vote for Republicans in down-ballot races, a spokesman for Mr Bush said.
"They voted 'None of the Above' for president," Freddy Ford said in an email about the couple, who now live in Dallas.
In a break from custom, neither Mr Bush, president from 2001 to 2009, nor his father, former Republican president George H W Bush, endorsed the Republican nominee for president, New York businessman Donald Trump.
Jeb Bush, the younger brother of George W Bush and the son of George H W Bush, ran against Mr Trump in the acrimonious and insult-laced Republican nominating contest this year.
- Reuters / BBC