The Catalan giants have endured a difficult start to the 2021/2022 season which has seen them win just four of their ten La Liga games.
A 1-0 defeat to Rayo Vallecano on Wednesday saw manager Ronald Koeman dismissed from the role, and has since been replaced on an interim basis by Sergi Barjuan.
The defeat has left Barcelona in ninth position, nine points behind leaders Real Sociedad.
Barcelona have also recorded just one win from their three UEFA Champions League games this season.
In spite of their struggles, Gaspart is confident Barcelona can bounce back and win the biggest titles this season.
"Barcelona will win the Champions League, La Liga, because there is still a lot of championship left, and the Copa del Rey, because it has not yet started," Gaspart, who was the club's president from 2000 to 2003, said to El Confidencial.
"Although one can be wrong, without optimism one is very unhappy. I have seen more difficult things in football. I hope that in the future we will aspire to the maximum, but I have already told you that it is never easy to win the great titles.
"If we don't succeed, we will have to qualify to compete in them next year."
Xavi Hernandez is the favorite to be named as new manager at the Camp Nou, and Gaspart is optimistic the former midfielder will do a good job if given the role.
He added: "Taking the results into account, was it premature for Guardiola, who had only trained the B team, to coach Barca? And Luis Enrique? Why won't it work [with Xavi]?
"Everyone was of the opinion that they had no experience and then the results came back. I wish him all the luck in the world because his luck is that of all Barcelona fans. He has shown for many years that being a Barcelona manager was one of his ambitions."
