The 58-year-old held the post from 2003 to 2010, overseeing Barca's resurgence as LaLiga's top side and a challenger in Europe.
Bartomeu and the board of directors resigned in October just days before a planned vote of no confidence in the president.
"Being president of Barca was an honor and gave me the best years of my life, and coming back to be president is the great challenge of my life," Laporta said on Monday.
"If the members believe in me, I'm convinced we'll get back to being the best. We'll do so with optimism, being faithful to ourselves.
"I want Barca to get back to being esteemed across the world, to recover the joy of being 'Barcelonista'."
The arrivals of Rijkaard and star forward Ronaldinho in the same year Laporta took over inspired the Catalans to success, as they claimed back-to-back league titles in 2004-05 and 2005-06, as well as the 2006 Champions League.
Laporta was also responsible for putting Guardiola in charge of the first team in 2008, a decision that yielded the club's first treble and the beginning of their most successful modern period.
Laporta, who was replaced by Sandro Rosell in 2010, previously stated he wanted to bring Guardiola back to the club if he ever returned as president, although the Manchester City boss this month signed a contract extension at the Etihad Stadium until 2023.
Speaking to TV3 in March, he said: "I'm working to introduce myself as a presidential candidate. I've been president before and I'm excited to be back.
"I would very much like Guardiola to come back but now he is at Manchester City and it is a decision that Pep should take. He is a benchmark for Barcelona and many Catalans would like him to coach Barca again. At the right time, I will speak to the person we think should be a Barca coach from 2021."