The 19-year-old has impressed in 21 league appearances this year and scored his sixth goal of the season during Wednesday's 2-2 draw against Fiorentina.
Zaniolo's form has earned him a reputation as one of the most exciting teenagers in Italy and has led to speculation linking him with a number of top European clubs.
He only moved to the Italian capital last summer from Inter for €4.5 million (£3.9m/$5.1m) and his father does not see him moving on again at the end of the current campaign.
"We have no preference. There will be discussions between clubs first. We will see in the summer. Now, Nicolo's priorities are the Champions League, and then the European Under-21 Championships."
Pressed on his son's future, Zaniolo's father added: "Are we renewing with Roma? Yes, exactly.
"Here, Nicolo is well, he is at a great club and he is happy with the club and his team-mates.
"Above all, the more experienced players, like [Edin] Dzeko, [Daniele] De Rossi and [Alessandro] Florenzi, are helping him a lot to grow.
"But, in the summer, despite the renewal with Roma, my son will be linked to so many teams. It's part of the game and it's normal."
Having been knocked out of the Champions League in a dramatic last-16 tie with Porto, and thrashed 7-1 in their Coppa Italia quarter-final against Fiorentina in January, Roma are set to end the 2018-19 season without a trophy as they sit seventh in Serie A.
But Zaniolo's father wants to see him win trophies in the future.
"I'd like to become a champion," he continued. "And I wouldn't mind seeing him play with the best, that is, with one of [Lionel] Messi, Cristiano Ronaldo, [Kylian] Mbappe or Neymar.
"But, above all, whether it's Roma or Los Angeles Galaxy, he hopes that in the coming years he will start winning trophies."