The Argentine forward wanted to leave last August before agreeing to stay and reconsider at the end of his existing deal, which expires on June 30.
The captain and record scorer's public show of discontent at Barcelona's leadership led to links with the likes of Manchester City and Paris Saint-Germain, who hoped to sign him on a free transfer.
However, the election of a new president in Laporta appears to have salvaged the 33-year-old's relationship with the club where he has spent the entirety of his professional career.
Amid reports a two-year contract extension is close to being agreed, Laporta has offered his most optimistic update yet.
He said: "Our talks with the agents of Leo Messi are going well. We hope they continue to progress. It's not done... but we know Leo wants to stay."
They are expected to be joined by Georginio Wijnaldum and Memphis Depay, whose respective contracts at Liverpool and Lyon are also up this summer.
This approach owes much to the club's latest financial results, which showed overall debt of almost €1.2billion.
However, Laporta insists Barca will still be able to renew Messi's deal, even as they look to make savings elsewhere.
He said: "The renewal of Messi does not depend on the audit. We are in talks with the agents of Leo and they are going well.
"We hope they continue to progress. What happens with Leo is not a matter of money, but a matter of the future and sports.
"He wants to continue with this idea because he wants a team that can help him win leagues, Champions [League] and titles in the short-term."