Trump said Obama released his birth certificate because of the pressure coming from Trump.
"When I got involved, I didn't fail," Trump said. "But I was the one that got him to produce the birth certificate. And I think I did a good job."
Trump accused Clinton of treating Obama with "terrible disrespect" when they were debating each other in 2008, when both were vying for the Democratic presidential nomination.
"You even sent out or your campaign sent out pictures of him in a certain garb, very famous pictures. I don't think you can deny that," Trump said.
In 2008, photos surfaced of Obama wearing Somali garb. Obama's campaign accused Clinton's team of dirty tricks; her campaign later denied it sent out the photo.
Clinton said Trump "has a long record of engaging in racist behavior," citing a 1970s racial bias case against Trump's company. The Republican nominee said it was settled with no admission of guilt.