In two posts on Twitter on Friday, Maltese Prime Minister Joseph Muscat said those on board the Ocean Viking will be transferred to Maltese navy boats outside the country's territorial waters.
French Interior Minister Christophe Castaner said that France would accept 150 of the passengers.
Malta's offer came after the NGOs running the Norwegian-flagged rescue ship - Doctors Without Borders (Medecins Sans Frontieres, or MSF) and SOS Mediterranee - warned the boat was running low on supplies and had only four days of standard food rations left.
The boat has been stranded at sea for two weeks, drifting between the Italian island of Linosa and Malta, keeping land out of sight for fear of agitating the traumatised passengers.
Many of the 356 migrants and refugees on board the Ocean Viking had fled desperate circumstances in their home countries and suffered horrific abuses in Libya, MSF said.
More than 100 of those on board were minors, about 90 of whom were unaccompanied, according to MSF.
In a statement on Friday, MSF Project Coordinator Jay Berger said the organisation was "relieved this long ordeal for the 356 people we have on board is finally over". But he questioned whether the "two weeks of excruciating wait" for people to be disembarked was necessary.
"While a coalition of countries have stepped up to give a humane response, European governments must stop these prolonged delays and ad hoc petty negotiations, and set up as a matter of urgency a disembarkation mechanism for people rescued at sea," he said.
SOS Mediterranee described the 14-day standoff as "unnecessary suffering" in a post on Twitter that was accompanied by a photo of passengers squeezed into cramped sleeping quarters on the deck of the ship.
"European states and their civil societies cannot allow such standoffs to happen again," it said in a statement.
The Ocean Viking will change crews and head back to Libyan waters for search and rescue missions, it added.
Libya is one of the main transit countries for migrants and refugees fleeing war and poverty as they try to reach Europe.