Italy has refused to allow the boat to dock, as it did last June when the Aquarius rescued 629 migrants off the Libyan coast.
This time it is carrying 25 people picked up on a small boat off Libya on Friday and 116 on a larger one later in the day.
The migrants are mostly from Somalia and Eritrea and about half of them are children, according to SOS Méditerranée, the NGO that operates the Aquarius.
Italy's far-right Interior Minister Matteo Salvini on Saturday declared that the Aquarius would "never see an Italian port" again and Spain, which allowed it to dock in June, has pointed out that it is not the nearest possible destination.
Corsica repeats offer
The president of the Corsican Assembly, Jean-Guy Talamoni, on Tuesday repeated the offer made in June to accept the Aquarius at one of the island's ports, as did the chairman of its executive committee, Gilles Simeoni.
Talamoni, like Simeoni a Corsican nationalist, called on the French authorities to "ensure that these people are accepted".
Simeoni, who chairs a committee of representatives of European islands, tweeted that he had written to the heads of the 23 local authorities in the group to urge them to demand a "real European policy for humanitarian aid in the Mediterranean".
Also on Tuesday Communist former transport minister Jean-Claude Gayssot, who now chairs the port authority of the southern French town of Sète, said he was ready to allow the boat to dock, while pointing out that he needed national clearance to do so.
Spain "saved Europe's honour" in June, he told RFI. "I believe that this time France should live up to its responsibilities and take an honourable stance, placing human solidarity above all else."
The French government appeared reluctant to give permission and had contacted other EU nations to find a destination for the NGO-run ship.
French right says go to Tunisia
The offer drew an angry response from Marine Le Pen, the leader of the far-right National Rally (RN, formerly the National Front).
"It's not up to the communist Jean-Claude Gayssot, president of the port of Sète, to decide our migration policy by accepting the Aquarius," she tweeted. "The government must at last send the signal of firmness that is required and close our ports to migrant ships!"
The RN on Tuesday called for Tunisia to take the Aquarius, a call echoed by the mainstream-right Republicans, both parties claiming that accepting its passengers would encourage people smuggling.
Gibraltar adds to pressure
France has accepted 78 of the migrants rescued by the Aquarius in June, 20 of the 87 rescued by another boat, the Open Arms, last week, 51 of the 233 on the Lifeline when it docked in Malta in June and 50 of the 450 picked up by the Italian navy in mid-July.
A further twist to the saga was added by the authorities in Gibraltar, who on Monday announced that it would stop the boat using its flag if it did not stop its rescue missions, on the grounds that it was registered as a research vessel.
SOS Méditerrannée slammed an "artificial difference" with "no technical basis".
On Tuesday afternoon Portugal to take some of the migrants aboard the Aquarius and the Spanish province of Catalonia offered to allow it to dock at one of its ports.