Twelve crew - 11 Chinese nationals and one Malaysian - were on the dredger. Seven of the Chinese nationals have been rescued and taken to hospital.
The tanker's 26 Indonesian crew members were not injured.
The dredger capsized and was partially submerged, while the tanker was damaged on one side, the Singapore Maritime Port Authority said in a statement.
Authorities are conducting rescue operations and have deployed search vessels and a helicopter.
They said shipping in the busy Singapore Strait was not disrupted.
The accident comes less than a month after a US destroyer and an oil tanker collided near Singapore, leaving 10 US crewmen dead.
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