Four people have been arrested in connection with the incidents, which took place in the city early on Saturday morning, South Yorkshire Police said.
In a series of three tweets, the force said: “We are dealing with two linked stabbing incidents in Sheffield city centre in the early hours of Saturday 30. Five people were injured.
“Four people have been arrested and detained in relation to the stabbing incidents in Sheffield early this morning.
“Roads are closed until further notice at Cambridge St, Carver St/Division St, Barker’s Pool and Bethell Wk leading to Backfields.”
A police spokeswoman said one person who was in a critical condition is now stable.
A police cordon remains in place and police vans are at the scene.
Beer barrels inside the cordon have what appears to be police riot shields placed on top, with blood on the surrounding pavement.
Forensics officers could be seen closely examining the shields and items within the cordon.
The scene is at the heart of Sheffield’s night life, and is close to the city’s main war memorial and John Lewis store.