Government sources told the BBC that the tribal fighters set fire to a building used by the militants.
The source said 10 IS militants and five tribesmen had died in the clashes that followed.
Issa al-Issawi, the exiled mayor of Fallujah, said that more tribesmen were joining the fight against IS.
Reports said the violence involved men from multiple tribes and IS members known as al-Hisba, who are responsible for enforcing religious strictures in the city.
"Clashes took place between sons of the al-Mahamda and al-Juraisat tribes against the al-Hisba group," Mr Issawi told AFP news agency.
He said the violence stemmed from tensions over increasingly difficult living conditions while Fallujah is besieged by Iraqi security forces.
Sheikh Majeed al-Juraisi, a leader in the al-Juraisat tribe, told AFP the clashes were part of an uprising against IS fighters in the city and called on the government and security forces to help.
Reports suggested the tribesmen had seized parts of the north of the city, but later ceded the ground back to IS jihadists.
Fallujah, which is about 50km (30 miles) west of Baghdad, has been held by anti-government forces since the beginning of 2014 and is now one of two Iraqi cities still under the IS group's control.
IS militants launched a sweeping offensive in June 2014 that overran large areas north and west of Baghdad, but security forces and allied fighters have pushed the jihadists back with support from US-led air strikes.
Local tribesmen have reportedly clashed with fighters from the so-called Islamic State (IS) in the IS stronghold of Fallujah in western Iraq.