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.

The Ghanaian midfielder signed a one-year deal after impressing on trial.
Buaben left Carlisle in May and joins ex-Dundee United team-mate Morgaro Gomis and fellow new signings James Keatings, Soufian El Hassnaoui, Neil Alexander and Alim Ozturk.
"It's great that Prince has put pen to paper," head coach Robbie Neilson told the official Hearts website.
"He's been at training for the last two weeks and he's played a few games for us.
"He's looking good and he's a top player. He's played in the SPL for a long time and he's played in England at a very high level so I'm pleased to get him.
"He gives us a bit of strength in there. We've got some good options now in the middle of the field. He's a good character as well.
"He hadn't done a pre-season after he finished up last season so I just wanted to see where he was and he's a fit guy.
"If this is the condition he's in, having only trained for just over a week, then I'm looking forward to seeing what we can get out of him in two or three weeks time."
Buaben moved to Dundee United in 2007 and was part of the side that won the Scottish Cup in 2010.
He also had a short spell on loan last season at Partick Thistle.

Concerns had been raised by the Pakistan Cricket Board over threats from Hindu extremists, who attacked the offices of Indian cricket's governing body last year.
However, PCB chairman Shaharyar Khan said on Thursday that the team had been cleared to play in March and April.
"I am pleased that our government has given its consent," said Shaharyar.
"As a duty of care, we have asked [world cricket governing body] the ICC to put in place special arrangements for the Pakistan cricket team while in India."
India and Pakistan have not played a Test series since 2007 due to fallout from the terrorist attacks on Mumbai in 2008, blamed on militants based across the border.
Since 2009, Pakistan have played their 'home' Test series in the UAE after a terrorist attack on the visiting Sri Lankan team led to a boycott on tours to the country.
The World T20 runs from 8 March to 3 April.
Pakistan are scheduled to play a qualifier team on 16 March in Kolkata, then hosts India on 19 March in Dharamsala.
Local tribesmen have reportedly clashed with fighters from the so-called Islamic State (IS) in the IS stronghold of Fallujah in western Iraq.
[NEXT_CONCEPT]
Prince Buaben has become Hearts' sixth summer signing after agreeing terms with the Tynecastle club.
[NEXT_CONCEPT]
Pakistan will play in the World Twenty20 after the government approved their participation in India.