The Argentina forward came close to joining Inter Milan this summer but the deal failed to materialise.
Palacio, who scored nine goals in 25 league appearances for the Grifoni last term, has since signed a one-year contract extension with Genoa that will see him remain at the club until 2014.
"I feared losing Palacio," admitted Genoa coach Alberto Malesani.
"He had received important offers that were difficult to turn down.
"But in the end our club president (Enrico Preziosi) listened to my advice and understood that Palacio had to remain at the club.
"The fact that Palacio stays is an important sign for Genoa."
Palacio will play alongside compatriot Lucas Pratto in Genoa's attack.
The Grifoni will be hoping to have a better start in Serie A than they did last season, where they only won one of their opening five games.
The Genoa fans will be hoping that under Malesani, who led Bologna to retain their Serie A status last term, the team will show more consistency.
Rivals Atalanta gained promotion to Italy's top flight this summer and are confident they can get off to a good start.
"We have worked very hard in recent weeks and we feel ready for Sunday's match at Genoa," said Atalanta goalkeeper Andrea Consigli.
"It's going to be a tough game but if we remain focused from start to finish, there is no reason why we cannot take home three points."
Atalanta coach Stefano Colantuono will have all of his players available with the exception of midfielder Ferreira Pinto, who is out with a knee injury.