The Argentina star announced in August his wish to leave Barca, his team since 2001, but he was forced to take a U-turn when the Catalan giants demanded his €700 million buyout clause instead of negotiating his departure.
Messi will become a free agent this summer with clubs able to approach him starting in January.
When asked whether Messi would have been released in the summer had the club's management commission, which he presides over, been in charge, Tusquets told Catalan radio RAC1: "Speaking from a financial standpoint, yes. But that is something that would have had to have been absolutely agreed with the coaching staff. From an economic standpoint, it would have been desirable to sell Messi."
Tusquets added that Messi's release, including the transfer fee and the money saved in wages, would have helped as "La Liga is demanding salary caps."
Josep Maria Bartomeu resigned as president of Barcelona on Oct. 27 and presidential elections are set for Jan. 24. Tusquets, meanwhile, spoke of the possibility of Neymar returning to the club.
Neymar told ESPN Argentina on Wednesday of his wish to play alongside Messi next season.
"If he comes as a free agent, we could consider it," Tusquets said. "If not, there is no money [for a transfer] unless there's a new president that holds a miracle in his hand. If we do sell [players] and the coaching staff is in agreement, then why not?"
Tusquets said the coronavirus pandemic has escalated the "worrying" financial situation at Barca.
"It's dreadful but there's hope. Once the stadium reopens, everything will change," Tusquets said.