Messi, 34, became a free agent for the first time in his professional career on Thursday as his €500 million ($594m) contract with Barca expired at the stroke of midnight on July 1.
La Liga president Javier Tebas has warned the Catalan club they must make cutbacks to be able to register any new Messi deal ahead of next season.
"We want him to stay and Leo wants to stay; everything is on track," Laporta told Onda Cero on Thursday.
"We still have the financial fair play [issue] to sort out ... we're in the process of finding the best solution for all parties."
Barca had the biggest salary cap in Spain prior to the pandemic at €671m ($796m) per year but, as of March, that had been almost halved to €347m ($412m).
The clear-out began last summer with the departures of Luis Suarez, Arturo Vidal and Ivan Rakitic, but Barca need to further reduce their wage bill to be able to register Messi's new deal and their four summer signings: Sergio Aguero, Eric Garcia, Memphis Depay and Emerson Royal.
So far, Barca have only been able to move on fringe players during the transfer window, with Juan Miranda, Jean-Clair Todibo, Matheus Fernandes and Konrad de la Fuente all departing. Left-back Junior Firpo is set to move to Leeds United, too.
Sources at the club say they expect things to speed up once Euro 2020 and the Copa America end, while they're also studying other ways to reduce the wage bill.
Any money Barca bring in from transfers could also help increase their salary cap for next season.
Messi's last four-year deal, which expired on June 30, was worth over €500m ($594m) combining signing on fees, his salary, bonuses and image rights.
Barca cannot afford to pay him at the same rate and are exploring ways to compensate him over a longer length of time to ensure he doesn't have to take too big a pay cut.
Options include continuing to pay him in an ambassadorial role if he chooses to leave Barca for Major League Soccer in a couple of years, as he's expressed he would like to, and the chance to return to the club after that in a backroom role.
Messi was first officially registered as a Barcelona player 20 years ago, in 2001, and since then he has always been contracted to the club.
He asked to leave last summer but was told no by then-president Josep Maria Bartomeu. Since then, Laporta has replaced Bartomeu and ESPN explained on Wednesday that Messi, as a result, has grown open to extending his stay at Camp Nou.
However, the significance of Messi becoming a free agent didn't go unnoticed, with former club Newell's Old Boys inviting him back on social media, while Brazilian minnows Ibis Sport Club promised him "the worst time in the world" if he joins them.