Antonio Tajani said that Humanity 1, a German-flagged vessel, would head for Catania, Sicily, and "be able to stay in our territorial waters for the time necessary for us to examine all the emergencies on board".
More than 1,000 migrants who were trying to reach Europe are currently aboard three rescue boats, and Italy has faced mounting pressure to let the humanitarian ships dock.
The charity rescue ships, including the Norwegian flag-bearing Ocean Viking and Geo Barents, have been at sea off Italy for more than a week waiting for permission from Rome to dock.
The Norwegian-flagged vessels have more than 800 people on board and are sailing off Sicily, while the German-flagged Humanity 1 has 179 people, including more than 100 unaccompanied minors and a seven-month-old baby with her mother.
"We will accept all those people, for example because they are minors, or because, according to what we know from the media, they are pregnant women or with young children, or people with fever", Mr Tajani said.
But he warned that "all those who do not meet these criteria will have to be removed from our territorial waters by the ship".
In an email statement to Reuters, ambassador Johan Vibe said Norway had “no responsibility” to take in the people on board two private Norwegian-flagged vessels.
The Norwegian ambassador’s response came after Italy’s new Prime Minister Georgia Meloni suggested that the countries under whose flags the ships are operating should take in the rescued migrants.
Italy sent letters last week to the embassies of Germany and Norway, saying non-government organisation (NGO) ships flying their flags were not following European security rules and were undermining the fight against illegal immigration.
"The primary responsibility for co-ordinating the work to ensure a safe port for those in distress at sea lies with the state responsible for the search-and-rescue area where such assistance has been rendered.
"Neighbouring coastal states also have a responsibility in such matters," Mr Vibe’s statement said.
The German embassy on Wednesday urged Italy to provide help swiftly, saying the NGO ships made an important contribution to saving lives at sea.
On Thursday, the charity SOS Mediterranee, which operates the Ocean Viking, said it had asked Greece, Spain and France.
French Interior Minister Gerald Darmanin told RMC-BFMTV that international law said Italy should take in the migrants, but also said that Paris and Berlin were ready to offer assistance.
"We have told our Italian friends, together with our German friends, that we are ready to take in, clearly as we have done in previous cases, some of the women and children so that Italy is not alone in receiving them," he said.
Search-and-rescue co-ordinator for SOS Humanity, Nicola Stalla, called the “blockade at sea” a “disgrace” and said delays in disembarking could have “life-threatening consequences”.
Petra Krischok, a press officer at the German NGO, SOS Humanity, who is aboard the vessel, posted on Twitter that the migrants were sleeping on the deck and could soon face rough seas after days of good weather.
In a video posted on the social media platform, a doctor on Humanity 1, said people aboard were getting sicker and suffering from skin problems, psychological stress; some showed signs of having experienced violence.
Earlier in the week, the Italian interior minister Matteo Piantedosi told the Corriere della Sera daily newspaper that Italy “cannot take in migrants who are picked up at sea by foreign ships operating without any planned co-ordination with the authorities”.
Migrant numbers have surged in Italy over the past week, with more than 6,200 people arriving since October 27 compared with 1,400 in the same period in 2021, according to government data.
The latest figures from the UN refugee agency show there have been just over 83,000 sea arrivals in Italy this year.