Celtic and Rangers have issued statements following allegations of sectarian behaviour during Saturday's derby game at Celtic Park.
Rangers said it was aware of "the disgust" felt by fans who "were subjected to a sickening and shameful display of outright sectarian hatred".
Celtic said they would not get involved in a "tit for tat spat" on the issue.
They went on to thank "the overwhelming majority" of fans for their backing of the team in their 5-1 win over Rangers.
Rangers fans group - Club 1872 - said banners which read "kill all Huns" and "know your place Hun scum" were threatening and disgusting.
Supporters were also offended by two blow-up dolls - one wearing a Rangers scarf, the other an Orange sash - which were dangled by ropes from the upper tier of the stadium.
Directors at the south side club said they shared the anger of their fans.
Internal matter
After the first Old Firm league derby in Glasgow in four years, it emerged that extensive damage had been caused to a toilet block at Celtic Park.
Police Scotland said it was aware of the destruction in the away section of the ground but the club was dealing with it as an internal matter.
Following the lunchtime game, the police said that the majority of Celtic and Rangers fans had behaved responsibly and "appeared to heed the advice given in the run-up to the fixture".
However, Assistant Chief Constable Bernard Higgins, of Police Scotland, added: "What is hugely disappointing and frustrating is that a minority of individuals behaved in a manner which has no place in our communities, football and modern society and which has resulted in Police Scotland carrying out investigations to identify those responsible."
An SPFL spokesman said: "The SPFL has a match delegate present at every Ladbrokes Premiership game. We will take time to study carefully the match delegate report from Saturday's Celtic v Rangers fixture."