The Brazilian, who joined the Liga champions from Santos for €57 million this summer, was diagnosed with a low red blood cell count - which can cause excessive levels of tiredness in players - following routine blood tests.
But Zubizarreta has insisted the 21-year-old's condition, which comes a month after the attacker had his tonsils removed, is something the club will manage and is not overly serious.
"We would have liked him to have been in perfect condition, but not many players arrive perfect for the pre-season," Zubizarreta said.
"We'll help make sure he's in the right condition. It's not a concern.
"There are quite a few days left before the league starts, although it is unlikely that he'll be 100 per cent adapted [to Barcelona's style] by then.
"But we can clearly see that he's quickly getting a grasp of our type of football."
Although the club are unfazed by the starlet's health, his father has revealed he is anxious over reports that his son has lost seven kilogrammes in weight since the beginning of July.
"Neymar has significantly low iron levels in his blood," Neymar da Silva Santos told TV Globo.
"The physical tiredness he has had is not normal and is quite worrying.
"As he has almost no fat to lose, losing that muscle mass is troubling."
Neymar made his Barcelona debut in a 2-2 friendly draw against Lechia Gdansk on Tuesday before appearing off the bench in an 8-0 win over Santos on Friday.
Barcelona begin the defence of their Spanish top flight title with a home fixture against Levante on August 18.