Summer Lee Seymour, 15, was killed on Friday on Haymarket at the junction with Castle Street.
A 17-year-old boy was injured and remains in a serious, but stable condition in hospital.
Melissa Hewitt, a bus driver who started the petition, has called the site "an accident waiting to happen".
First South Yorkshire said there were no passengers on the bus at the time.
The company said the driver had not been suspended, but "temporarily relieved of his duties".
Ms Hewitt said: "It was heartbreaking, it really saddened me and I wanted to make a change so we don't lose any more lives.
"Only yesterday I had somebody walk out in front of my bus and I had to slam my brakes on."
A crossing did already exist about 20m (65ft) up the road, but she said another was required in the area due to the consistent number of pedestrians.
She said she was "overwhelmed" with the public's response to the petition so far and hoped the situation would be debated by Sheffield City Council.
A council spokesman said: "We are co-operating with the police investigation into this accident, but until the outcome of this investigation has been completed, it would be premature to announce any changes that could be introduced in that area of the city centre."
More than 4,000 people have signed a petition calling for a pedestrian crossing where a teenage girl died after being hit by a bus in Sheffield.