The incident was captured by a shocked woman on her mobile phone as a row between bus passengers escalated.
A disagreement broke out between two women on a number 83 bus, operated by First, on Arundel Gate on Wednesday afternoon.
It is understood to have centred around the number of parents on the bus with children in pushchairs.
The bus driver intervened in a bid to restore calm but during the incident one mum with a young child spat at an older woman on the bus.
The woman who captured the video and shared it on Facebook said: “This is not what you expect to see on a busy bus in the city centre. It was shocking.
“I was walking past the bus when I heard screaming and shouting so I stopped to see what was happening and there was a woman on the pavement with a pushchair arguing with an older woman on the bus.
“People were telling her to calm down and some passengers were so fed up of waiting for the bus to set off again, they got off and walked away.
“It was over the number of pushchairs on the bus. I saw one woman, with tears in her eyes, get off the bus with her pushchair while it was all going on.
“I was really worried about what was going to happen especially because children were there while it was all happening.
“There was lots of aggression and swearing and then the younger woman spat at the older woman. It wasn’t nice to witness.
“Another bus driver nearby was asked if he was calling the police, but he said he didn’t think the could come out to it because they are really busy.”
The woman accused of spitting denied the act, and claimed the row broke out when she boarded the bus with a friend, who also had a pushchair, and then a third mum with a buggy boarded after them.
She said the third mum refused to fold her buggy down and ‘rammed’ it into her leg.
The woman said an older woman on the bus got involved and her hair was pulled during the incident.
The incident is being investigated by South Yorkshire Police.
“We received a call on Wednesday, October 2. The caller reported a hate crime incident that occurred while on a bus on Arundel Gate in Sheffield.
“Enquiries are on-going and anyone who witnessed what happened are asked to call us.”
First South Yorkshire said: “Our 1,000 drivers work hard across South Yorkshire to deliver the best service they can, sometimes in difficult circumstances.
“We support our staff through a range of training and development opportunities including our ‘Journey Makers’ programme which enables drivers to discuss and share experiences to help deliver best practice resolution.
“Although such incidents as outlined on the video are rare, the safety of our staff and customers is paramount and where issues do occur, we will take action with partners such as South Yorkshire Police where necessary.”
Call South Yorkshire Police on 101 and quote incident 646 of October 2.