Open Arms had repeatedly asked Italian authorities for permission to bring the migrants ashore, but met resistance due to Italy's ban on private rescue ships.
Earlier, 10 migrants jumped off the ship, hoping to swim to the island.
Some people have been on board the Open Arms for 18 days, putting them on track to surpass the record of 19 days which 32 migrants spent stranded on another charity rescue ship, the Sea-Watch 3, in January.
At least 15 more migrants jumped into the sea Tuesday from the Open Arms rescue ship in desperate bids to reach the shores of Italy after 19 days on the boat in deteriorating conditions as Italy refuses to open its ports.
Open Arms described the situation on board as "out of control" and "desperate".
The current showdown between the NGO and Interior Minister Salvini is just the latest case of a foreign-based migrant rescue NGO challenging the closed port policy of the Italian government, a policy that has led to a massive reduction in drowning deaths since 2018.
"Being firm is the only way to stop Italy from becoming Europe's refugee camp again", he tweeted.
The standoff has fuelled Salvini's campaign against migrant boats from Africa, and comes as he is trying to drag Italy into snap elections.
Those who jumped into the sea were not returned to the Open Arms but taken ashore by the Italian coastguard, where doctors were waiting for them. The ruling coalition, in danger of collapse later on Tuesday, has split over the issue.
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"They are seeing the coast at 200 metres, and they see they can not reach the coast so they feel like they are in jail", Open Arms spokeswoman Laura Lanuza had told Al Jazeera.
Salvini's political rival, Transport Minister Danilo Toninelli, who controls the Coast Guard but not port access, has offered to take the migrants on a Coast Guard vessel to Spain, which has offered them safe harbour.
The Spanish government said it had not received a clear answer from Open Arms on how it would like to proceed.
Without a flag, it would be hard for the ship to continue running rescue missions.
"[But] the NGO has incredibly refused [this offer], with an attitude that makes one suspect there is bad faith on their part", Toninelli said.
"We offered them everything because we don't want these lives to be in danger any longer", she told news radio Cadena Ser.
Open Arms chief Oscar Camps said Saturday that tensions were rising and fights breaking out that threatened the safety of all on board.
Six European Union countries - France, Germany, Romania, Portugal, Spain and Luxembourg - have offered to take in the migrants aboard the Open arms boat.
The head of the mission on the Open Arms ship, Anabel Montes, now insists on disembarking the 98 people on board in Lampedusa in Italy later today.