Some Democrats urged the college – a group of 538 electors chosen by state parties – to change their votes to Clinton after she won the popular vote by a margin of nearly 2.9m ballots, Reuters reported.
The Electoral College vote is typically a formality, but this year Clinton's popular vote win and allegations of Russian interference in the election have caused protestors around the country to call for electors to dump Trump.
Despite the lobbying, there is no sign that electors will change the outcome of the election, which Trump won with a majority of 306 Electoral College votes.
A candidate needs 270 votes to win the presidency.
At least five Democrats who were committed to back Clinton have cast ballots for others, however, in the largest number of vote swaps in more than a century.
By late afternoon, no Republican elector was reported to have cast a ballot for anyone other than Trump, although one Texan elector had written that he planned to.
The Electoral College votes will be officially counted on 6 January, but there is almost no chance the vote today will change the outcome of the election. Trump is set to take the White House on 20 January.
"I don't think you have to wait for a surprise on January 6," Robert Erikson, a political scientist at Columbia University, told Reuters.
"If it's in the works, it sure has been a secret plot," Erikson said.
The Electoral College was established in 1787 and assigns each state electors equal to its number of representatives and senators in Congress.
When voters go to the polls to cast a ballot for president, they are actually choosing a presidential candidate's preferred slate of electors for their state.