All six fires began outside, starting on piles of debris, San Jose Fire Capt. Don Torres said.
The first three fires were all reported between 1:57 a.m. and 2:11 a.m. in the area around Clark Street and Dorothy Avenue in Willow Glen, but did not cause much damage.
Three more fires were reported between 4:38 a.m. and 5:01 a.m. in the area around Willow and Vine streets, about a mile away in the Tamien neighborhood.
The second of those fires reported at 4:49 a.m., in a debris pile behind a convenience store in an area containing the occupants' garbage and storage, Torres said.
RELATED: Police say Contra Costa County car fires were arson and likely linked
That fire spread to the store in the 200 block of Willow Street, leading to a two-alarm response. An aggressive interior attack by firefighters contained it to the store, fire officials said.
Last month, San Jose fire officials called in federal authorities to investigate a fire at the Assyrian Church of the East in Willow Glen.
Two or three other fires had been reported in the neighborhood that same week, fire officials said May 8.