The crash happened in Cook Street in Glasgow, near the O2 venue, at 11.35 BST.
The injured people were taken to the Queen Elizabeth University Hospital.
A Scottish Fire and Rescue Service spokesperson said: "A number of additional casualties have been treated at the scene but do not require to be taken to hospital."
Ch Insp Elaine Tomlinson, of Police Scotland's Greater Glasgow Division, said: "Around 11.35am on Sunday, 21 May, 2023, we received a report of a bus crashing into a bridge on Cook Street, Glasgow.
"Emergency services attended and 10 people were taken to various hospitals for treatment.
"A number of road closures remain in place, with disruption to some rail services.
"I would like to thank the public for their co-operation and ask they continue to avoid the area while enquires are ongoing."
First Bus confirmed it was one of its buses which was involved.
A company spokesperson said: "We are working closely with the emergency services at the scene and our thoughts are with those involved in the incident."
Network Rail said its engineers would need to carry out an inspection of the bridge before it could be used by trains again.
The rail infrastructure agency said: "We're assisting the emergency services, who are responding to a bus trapped beneath a bridge between Glasgow Central and Paisley Gilmour Street.
"We need to complete a safety inspection of the bridge before trains can use it again. We can only do this once the bus has been removed."