Now Inter Miami, Messi is at odds with Laporta and his board over the way he was pushed out of the club.
The Argentine made a return to the new stadium last month - without the club's and Laporta's knowledge.
However, the president remains confident of fixing the rift, stating during a Barca function this week: "People find common ground when they talk.
"As for Leo Messi, we plan—when the time is right for everyone—to pay him the greatest tribute and erect a statue. It would be even more extraordinary in a stadium filled with 105,000 fans.
"He was the best player, perhaps in the history of football, and certainly at Barcelona. But I don't want to stir up controversy or speculate on unrealistic things.
"He has a contract with another club. We discussed the possibility of naming the stadium in his honour in the board of directors, but we decided it was inappropriate, as it's usually done for people who are no longer with us."