The 38-year-old Thompson's comments were made during a brunch Sunday hosted by the Kenosha County Republican Party and attended by Republican National Committee Chairman Reince Priebus and Wisconsin Republican Party Chairman Brad Courtney, the Milwaukee Journal Sentinel reported.
The video was taken by a Democratic Party operative and posted online.
Jason Thompson has been involved in his father's campaign for an open U.S. Senate seat against Democratic Rep. Tammy Baldwin.
"We have the opportunity to send President Obama back to Chicago -- or Kenya," Jason Thompson, an attorney, was videotaped as saying.
His comments prompted laughter, with one woman saying, "We are taking donations for that Kenya trip," the Journal Sentinel said.
A group of Obama critics promotes the discredited "birther" argument that he was not born in the United States. Obama's father was Kenyan and the White House released the president's birth certificate last year showing he was born in a Hawaii hospital.
Asked about his son's remarks, the former governor said, "I don't know what you're talking about."
Thompson's staff later sent an email on the controversy, saying, "The governor has addressed this with his son, just like any father would do. Jason Thompson said something he should not have, and he apologizes."
Priebus and Courtney did not return calls, the Journal Sentinel said.
The former governor backed away from his own remarks Sunday after he called his opponent "anti-Jewish" when commenting on allegations she flip-flopped on Iran sanctions.
Later he clarified that he meant to say Baldwin was "anti-Israel."
Charlize Theron not engaged to Sean Penn 'yet'
Pregnant Mila Kunis wins 'Best Villain' at MTV Movie Awards