Gattuso's side face Estonia in Tallinn on Saturday before returning home to meet Israel at Stadio Friuli in the northern Italian city of Udine the following Tuesday.
"We need to work with concentration and not get caught up in everything that's coming from outside. The atmosphere is certainly not great, but we need to focus on doing our part," Gattuso told reporters on Tuesday.
The match in Udine will take place under heightened security and with a reduced crowd, following demonstrations in Italy in recent weeks against Israel's offensive in Gaza.
There were large pro-Palestinian demonstrations across the country last week.
"We're going to Udine and we know there won't be many fans: I understand that, I understand the concerns," Gattuso said.
"We also know that we have to play, otherwise we’ll be given an automatic 3-0 defeat."
Israel attacked Gaza after a Hamas-led attack in 2023 in which some 1,200 people were killed and 251 taken hostage, according to Israeli tallies.
Palestinian health authorities say Israel's two-year ground and air campaign against Hamas militants in the Gaza Strip has killed more than 67,000 people, with nearly a third of the dead under the age of 18.
"It pains the heart to see what is happening," added Gattuso. "We know full well there will be 10,000 people outside the stadium and 6,000 inside."
Norway are top of Group I with 15 points ahead of Italy and Israel on nine, although the Italians have a game in hand.
Gattuso has made a strong start as Italy coach with two wins from two, though the 5-4 victory over Israel in Hungary in September was anything but straightforward as a last-minute goal from Sandro Tonali secured the win.
"There are many things we need to fix. It’s not just about the defence, but about being able to sniff out danger in all areas of the pitch," he said.
"At the same time, there are also many positives. We were the ones who allowed Israel back into the game and we were very lucky to take the result home. We’ve got to improve our attitude, it’s not about the tactical system."