Libyan coast guards rescued about 85 migrants off the shore east of Tripoli on Saturday but about 40 more migrants were believed to be missing, a coast guard officer said.
The migrants were rescued about six miles (10 km) north-west of the town of Garabulli after fishing boats spotted them at sea, said Muammar Mohamed Milad, a local coast guard official.
"Due to the heavy load on the rubber boat, the wooden base got broken and it started sinking," said Milad.
"According to the survivors about 40 others are missing, including seven children," he said. The body of one woman was had already been recovered.
Garabulli, about 50 km (30 miles) east of the capital, is a common departure point for migrants trying to reach Europe. Smugglers usually cram the migrants into flimsy rubber boats with homemade wooden bases.
Most make it to international waters where they are picked up by ships and taken to Italy, but some are intercepted by the Libyan coast guard, or break down or sink in Libyan waters.
The vast majority of migrants trying to cross to Europe by sea use the central Mediterranean route between Libya and Italy.
More than 85,000 migrants have arrived in Italy so far this year. Some 2,150 are known to have died attempting the crossing over the same period, according to the International Organization for Migration (IOM).