Emergency services provisionally put the death toll at seven earlier on Friday but this rose mid-morning to 10.
Prime Minister Mario Draghi confirmed the toll, while saying it could change, before heading to the town of Ostra near Ancona, one of the places worst hit.
More than 400 millimetres (16 inches) of rain fell there over a few hours on Thursday evening.
One of the four people reported missing was a child travelling in a car. The mother was rescued but the child was washed away by the floodwaters, AGI said.
Water swept through towns and villages, turning streets into rivers and flooding homes, with the worst damage reported in the Ancona area.
All the victims reported so far are in the province of Ancona, according to local media.
The fire brigade said it had rescued dozens of people overnight “who took refuge on roofs and in trees” during the floods.
#Maltempo #Ancona, 180 #vigilidelfuoco al lavoro: salvate nella notte decine di persone rifugiatesi sui tetti delle abitazioni e sugli alberi.
Più di 150 gli interventi effettuati, 7 persone decedute e 3 risultano disperse.
Nel filmato il lavoro a #Senigallia (AN) #16settembre pic.twitter.com/9KuzHkkFGr
— Vigili del Fuoco (@vigilidelfuoco) September 16, 2022
Another tweet from the fire brigade read: “the search for missing persons continues among mud and felled trees”.
The wave of bad weather that hit the area “was not expected at these levels, we had no alert in place. The flooding was sudden,” Marche regional councilor for civil protection Stefano Aguzzi told Ansa.
“There was no time to intervene. There are people who may have been on the street or who had gone out not realising the danger.”
Several areas in Ancona were without electricity or telephone connections. Schools were closed on Friday in the affected areas.
Draghi announced five million euros in emergency funds for the area and visited Ostra to offer his condolences in person.
“We will do everything possible… you can count on us,” said the premier, who is set to leave office following elections on September 25th.
Speaking in Rome earlier, Draghi also made an explicit link between the flooding and global warming, saying: “We see it concretely in what happened today how the fight against climate change is fundamental.”
Francesco Rocca, president of the International Federation of the Red Cross and Red Crescent Societies, said its teams were heading to help.
“Very concerned by the growth of extreme weather events,” he said on Twitter.