Tunisia's Defence Ministry said the boat had sunk off Sfax, on Tunisia's southeast coast while on its way to Europe.
Ministry spokesman Mohamed Zekri said navy units were sent to search the water for those missing.
Around 33 others were rescued by workers from an offshore oil platform near the North African country.
Flavio Di Giacomo, a spokesperson for the Mediterranean coordination office of the International Organisation for Migration (IOM), said that the survivors were all from Bangladesh.
The boat had departed from Zwara in Libya on Sunday, Di Giacomo said on Twitter.
An IOM spokesperson in Tunisia, Riadh Kadhi, said the survivors had reported that the boat carried about 90 passengers when it left Libya.
Safa Msehli, an IOM representative in Geneva, tweeted that teams from the United Nations migration agency were providing humanitarian assistance and shelter.
Libya is a frequent departure point for Europe-bound migrants making the dangerous Mediterranean Sea crossing.
On Monday, the Tunisian navy rescued another 113 migrants from Bangladesh, Morocco and sub-Saharan Africa as their boat was about to sink off the island of Djerba.
Meanwhile, numerous boats were intercepted by the Libyan coastguard and taken back to Libya on Sunday night.
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