He uses Obama's full name, Barack Hussein Obama, and said: "In many respects, you know, they honour President Obama.
"He is the founder of Isis (another name for IS)."
Trump, who says he is an evangelical Christian, has also accused his Democratic rival, Hillary Clinton, of founding the militant group, in the past.
Mr Obama, who has also professed a Christian faith, has served two terms as leader of the United States.
Mr Trump has previously questioned his and Hillary Clinton's faith and has long blamed Mr Obama and his former secretary of state Ms Clinton for pursuing Middle East policies that created a power vacuum in Iraq which was exploited by IS.
The White House has not commented on Mr Trump's accusation.
