The missing include four Chinese nationals and one Malaysian, the MPA said. Seven other Chinese nationals have been rescued, it said.
The dredger capsized and is currently partially submerged, while the tanker reported damage to her starboard, MPA said.
Many dead as boat capsizes off Malaysia
Seven sailors missing after US Navy destroyer collision
Singapore authorities are conducting search and rescue operations and the Singapore Navy has deployed three patrol craft. The Singapore Air Force has also deployed a helicopter to conduct an aerial search, the MPA said.
It said there had been no disruption to shipping traffic in the Singapore Strait.