As night fell in Lampedusa, the crew and passengers of the Alex remained aboard, watched by police, the BBC reported on Saturday.
Despite warnings, it sailed to the port on Saturday because of "intolerable hygienic conditions" aboard.
Matteo Salvini, Italy's Interior Minister, has said again he will not allow migrants to disembark.
"I saw that the German government wrote me a letter today asking to reopen the ports and to let all the migrants disembark," he said on a Facebook live stream. "No, no, no, absolutely not."
Last year, he closed Italian ports to rescue ships and Italy has introduced fines for anyone sailing into its waters without permission.
The Alex is run by the Mediterranea charity, which tweeted that its exhausted crew were living through a "surreal situation", and prolonging the wait was "an unnecessary cruelty".
"It's an unbelievable situation," spokeswoman Alessandra Sciurba said at the quayside. "There are people at risk of fainting, the toilets are not working. It's as though we are kidnapped."
"Nothing is happening about disembarkation and nobody knows what will happen."
Meanwhile, another NGO ship, the Alan Kurdi - operated by German charity Sea-Eye - is afloat in international waters just outside Lampedusa, carrying another 65 people.
Italy has been one of the main destinations for migrants attempting to reach Europe via the North Africa route, mostly from Libya.
Salvini, from the right-wing League party, takes a hardline stance against migrants and rescue ships - something that has seen both his and his party's popularity increase.
In February, it was reported that Italy had rejected a record 24,800 asylum applications between October 2018 and January.
And a poll published in the Italian newspaper Corriere Della Sera on Saturday said 59 per cent of Ital approved of Salvini shutting Italy's ports off to NGO-run vessels.
( With inputs from IANS )