The bus, with seating on the upper deck for sightseeing, was too high to pass under the Alexandre III bridge and scraped the roof of the tunnel before stopping after a few metres (yards).
The Paris fire service said a woman who was among those hurt had serious injuries.
The crash came during a large-scale event on the Seine to promote the French capital's bid to host the 2024 Olympics, with a floating running track installed across the river just next to the bridge.
A police source said the bus was not travelling along its usual route, having been diverted because of the Olympic event.
The Big Bus Tours company, which offers sightseeing trips around Paris on open-top buses, confirmed to AFP that one of its vehicles had been involved in an accident, without giving further details.
Traffic was cut off in both directions to allow the injured to be treated and the bus to be towed away.
The ornate Alexandre III bridge over the River Seine is a magnet for tourists, especially honeymooners keen to take selfies with the Eiffel Tower in the background.
In 2008, a six-year-old boy and a 45-year-old man died after a tourist pleasure boat carrying 12 people sank in the Seine after hitting a bridge.
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