Bari, who last played in Serie A in 2010-11, finished seventh in Serie B last season and qualified for the promotion playoffs.
But they were refused a licence last month for next season and forced to restart in Serie D, the highest of the amateur leagues.
On Tuesday, the city's mayor Antonio Decaro offered the ownership of the club to De Laurentiis ahead of 10 other candidates.
"This club has a very long history of 110 years. We must rise quickly and try to return to Serie A," De Laurentiis told a news conference on Wednesday.
"One thing is certain. We cannot mix Naples and Bari, two different cities, each with its own history and its own fans that must be respected... Bari will not just be an appendix to Napoli."
De Laurentiis said his son Luigi would be responsible for the club.
"I have finally convinced him after several days of phone calls and discussions, because he was never interested in football up to now," he said.
He added: "We must try to change the rules that do not allow the same owner to have two teams in Serie A -- it seems to be impossible now but in the future it should become normal."
It is common for Italian clubs to be relegated and re-founded following financial crisis.
De Laurentiis himself bought Napoli in 2004 after they were demoted to Serie C after going bankrupt.
Fiorentina have also been re-founded as were Parma, who were relegated to Serie D in 2015 after going bankrupt and will return to the top flight this season after three successive promotions.
(Writing by Brian Homewood; Editing by Christian Radnedge)
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