The Air Quality Index reading near Berkeley has reached as high as 248, a level at which "serious consideration should be given to rescheduling the activity or moving it indoors," according to the NCAA.
"Prolonged exposure and heavy exertion" at that level, the NCAA says, "should be avoided."
In a release announcing the cancellation, Cal said a significant amount of smoke infiltrated Haas Pavilion, where the Bears play.
There has been no firm decision regarding Saturday's football game against Stanford in Berkeley.
"Cal is continuing to monitor air quality, including reviewing data provided by the Bay Area Air Quality Management District (BAAQMD) and staying in regular communication with officials from Pac-12 and Stanford," the school said.
An NCAA spokesman said the final decision on Saturday's Cal-Stanford football game, known as "The Big Game," will be made by the respective schools and the conference.
"They also have the ability to make more conservative decisions than NCAA guidance outlines," the spokesman said.
Poor air quality in the area has also forced the University of San Francisco to cancel Friday night's men's basketball game against Arizona State. USF said it would work with the Sun Devils to reschedule the game.
The FCS game set for Saturday between Sacramento State and UC Davis, in Davis, is also in jeopardy. Davis' campus will be closed Friday, and athletic officials are expected to come to a decision about whether the game will be played by 10 a.m. Friday, according to a spokesman.