Messi, 33, is out of contract at the end of the month, but reports Tuesday suggested he had given the green light to a new two-year deal at Camp Nou.
"Talks are ongoing with Messi's people," Laporta said in a news conference when asked whether an agreement was in place. "As I have said, they're going well. We hope to keep making progress.
"It's not done but I am convinced, like [Aguero] said, that they would like to play together here. I have always said the issue with Leo is not financial for him.
"[Messi] is showing a lot of understanding and wants to stay and have a competitive team that wins trophies. The sooner we have a decision, the better, of course, but we're not in a rush."
If Messi signs a new deal, it will mark a remarkable turnaround from last August, when he informed the club of his desire to leave following an 8-2 defeat to Bayern Munich.
Since then, Laporta has returned as president, replacing Josep Maria Bartomeu, and has promised to improve the team to help Barca compete for Europe's biggest trophies once again.
"We lost the league and there will be consequences," Laporta added in reference to overhauling the squad. "That's the level of demand there has to be here.
"We want to improve the team and there will be more [signings]. We're working on it. We need to balance the squad, and you will see what we're working on soon. Next week there will be more news."
Laporta also gave the biggest indication yet that Ronald Koeman will continue as coach.
Sources told ESPN last week that Barca were exploring possible replacements but that a lack of credible alternatives could see the Dutchman kept on for the second year of his contract.
"I would say he's closer to [staying] than [leaving]," Laporta said. "We're in a period of reflection, but talks are going well."