The Malta-based, Migrant Offshore Aid Station, said it co-ordinated the operation along with Italian, Irish and German ships.
Italy's coastguard could not confirm the numbers, but said it was trying to help around a dozen other vessels.
HMS Bulwark, a British Royal Navy ship, is also making its way towards Libya to assist with the situation.
BBC correspondent Jonathan Beale, who is on board, said the ship's crew would launch helicopters and landing craft on Sunday morning in the expectation they would pick up more migrants.
The Bulwark has already saved almost 1,800 people over the past month.
Italian navy ship Driade rescued 560 of the migrants on Saturday, including women and children, while the Irish vessel LE Eithne picked up 310 people.
The numbers of migrants attempting to cross the Mediterranean in the hope of reaching Europe, has increased by more than 10% in the first five months of 2015.
Italy's government predicts a total of 200,000 will arrive on its shores this year, up from 170,000 in 2014.