The news comes a day after three babies were found dead and 100 more went missing in a shipwreck apparently caused by an explosion on board a boat that had set off from Libya.
It also follows a controversial deal reached by EU member states to stem the influx of migrants. The accord struck on Friday includes the setting up of secure centres for migrants in the bloc, "disembarkation platforms" outside the bloc and sharing out refugees among member states.
Spanish NGO Proactiva Open Arms said on Twitter the 59 migrants were now on board their ship "heading for a secure port".
MSF says migration deal will 'block people' at Europe's doorstep
"Despite the hurdles, we continue to protect the right to life of invisible people. Their stories on what they experienced (in) Libya are terrifying."
Italy's hardline Interior Minister Matteo Salvini said on Facebook that the NGO's boat had reached the migrants' vessel before Libyan coastguards inside Libya's rescue operation zone.
"The nearest port is Malta, the organisation and flag are Spanish: they can forget about coming to an Italian port," he added.
Malta's interior minister retorted on Twitter that the migrants had been rescued between Libya and Italy's Lampedusa.
"Stop spreading incorrect information by pointing the finger at Malta without any reason," Michael Farrugia said.
It is unclear where the vessel will anchor.
Italy's new populist government and Malta earlier this month refused to let dock French NGO rescue ship Aquarius carrying 630 migrants. The move triggered an international outcry before Spain stepped in to help.
Malta then let another charity rescue boat, Lifeline, dock with 230 migrants aboard.
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