European navies and nongovernmental organizations are struggling to locate and rescue migrants from the rickety ships, and thousands may be pulled to safety Sunday, a spokesman for the organization told CNN. "The numbers are high, and they are rising," Federico Soda said.
The warning comes after nearly 3,500 migrants were rescued Saturday.
The HMS Bulwark, a British Royal Navy ship, has rescued more than 1,000 migrants on Sunday alone, including 10 pregnant women, a spokeswoman for the UK Defence Ministry told CNN.
It is the largest rescue operation the Bulwark has had since deploying to the Mediterranean on May 5, and the numbers are climbing as the operation is still ongoing, she said.
The ship has rescued more than 2,700 migrants since it was sent to the Mediterranean.
A spokesman for the U.N. refugee agency, UNHCR, said naval ships from Italy and Spain were also involved in the effort to rescue migrants on Sunday, along with the Italian coast guard.
The Italian coast guard has received requests for help from 14 vessels in distress, carrying an estimated 1,500 refugees and migrants, the UNHCR's William Spindler said.
They have rescued migrants from 11 vessels, and operations to find the other three boats and rescue those on board continue.
Commercial vessels are also being directed to the rescue area to help with the operation, Spindler said.
A spokesman for the Italian coast guard, Andrea Sbalbi, earlier said 10 rescue operations were ongoing for migrants at sea in wooden fishing boats and rubber dinghies.
Migrants taken to Sicily
Two German naval ships that coordinated Saturday's rescue of thousands of migrants north of the Libyan coast are on their way to Sicily to drop them off, a spokesman for Germany's Bundeswehr Joint Forces Operation Command told CNN on Sunday.
The two ships, the Berlin and the Hessen, picked up 1,411 migrants from four vessels Saturday -- 939 men, 327 women and 145 children. In just one vessel there were 563 migrants, the spokesman said.
The Hessen was expected to dock at Palermo and the Berlin at Trapani on Sunday. The ships will be replaced early next week by two new ships from the German navy that are on their way to the Mediterranean.
The Irish navy ship LE Eithne, which was involved in Saturday's rescue operation, is returning to Italy with 399 migrants on board -- 280 men, 78 women and 41 children, a spokeswoman for the Irish Defence Forces told CNN on Sunday. The ship is not involved in Sunday's ongoing rescue operation in the Mediterranean.
According to the Italian coast guard, the ship is due to bring the migrants to Taranto, in Italy's Puglia region, early Monday.
A dozen ships from several countries, along with at least one private ship, were involved in the rescue operations Saturday, the Italian coast guard said.
Ian Ruggier, a member of the humanitarian group Migrant Offshore Aid Station, was on one of the rescue boats.
Some rescue ships turned up with rescued migrants already on board, according to Ruggier. That's what happened with one German naval vessel that picked up 301 migrants from an unreported vessel as the German ship was on the way to the larger rescue scene, a spokesman for Bundeswehr Joint Forces Operations Command said.
The German ships went to the main site and found seven packed boats -- far more than they expected -- according to the spokesman for the command, which is part of Germany's military.
The Italian Interior Ministry reported that several of its ships were involved, including from its coast guard and navy. Some Italian rescue vessels reached the island of Lampedusa on Saturday night.
Europe dealing with immigration crisis
The prospect of migrants being in jeopardy in the Mediterranean is serious but hardly surprising. People from impoverished and conflict-torn countries such as Syria, Somalia, Sudan and Eritrea have taken to the sea to reach Europe in especially large numbers of late, often putting their fate in the hands of people-smugglers.
Many times they haven't made it. Hundreds of migrants died in April when their crammed 66-foot (20-meter) boat capsized roughly 70 miles (113 kilometers) north of Libya. That got a lot of news coverage, but it didn't stop the carnage -- including dozens feared drowned after falling into the Mediterranean last month as a rescue vessel approached, according to the aid organization Save the Children.
The United Nations estimates that, as of the end of May, 90,000 refugees and migrants had crossed the Mediterranean into Europe so far this year. Just over half landed in Italy, with roughly 42,000 in Greece and the rest recorded in places such as Spain and Malta. The UNHCR's Spindler said that about 1,850 had died or were missing at sea.