Laporta said Neymar approached Barca about coming back and that at the time the Camp Nou president had a different understanding of the LaLiga fair play rules which led to a transfer window of cutbacks at the club.
"We tried to sign Neymar," Laporta told TV3 on Monday, confirming reports that had appeared in Spain earlier in the day.
"At that moment [after being elected in March], it seemed like an interesting signing. He got in contact. He was crazy about coming.
"We interpreted fair play in another way at the time. If we had known all the rules [in LaLiga], we would not have made that offer."
Laporta replaced Josep Maria Bartomeu as president earlier this year and inherited the club in a financial crisis, with gross debt reaching €1.35 billion.
Barca's wage bill equated for 110% of their revenue before Messi left and, even following the exits of Messi and Antoine Griezmann, it remains at 80%, Laporta confirmed on Monday. LaLiga's rules compel clubs to keep that figure at around 70%.
PSG, meanwhile, aren't limited by such rules in France and have been able to add Messi to a squad that already includes Neymar and Kylian Mbappe, as well as signing Gianluigi Donnarumma, Sergio Ramos, Achraf Hakimi and Georginio Wijnaldum this summer.
"PSG have broken all the fair play rules that there are," Laporta said.
Laporta also branded reports Barca tried to sell Ansu Fati this summer as "lies" and said talks are underway with Sergi Roberto and Samuel Umtiti, among others, for further pay cuts. Gerard Pique, Jordi Alba and Sergio Busquets have already had their contracts adjusted.
On Griezmann, who joined former club Atletico Madrid on loan on deadline day, Laporta said he gave his all "but didn't fit" the team's style of play, confirming that the €40 million option to make the loan permanent becomes obligatory if the Frenchman plays at least 50% of games for Atletico.