The migrants were rescued in six different operations off Libyan shores on Saturday morning and their boat awaits confirmation for somewhere to dock.
An Italian official says the country will turn away a boat full of more than 600 migrants from its shores and direct them to Malta instead.
The move is reportedly part of new Interior Minister Matteo Salvini's mission to halt the flow of incoming migrants, following up on his previous electoral promises.
Responding to Italy's comments, Malta said it had nothing to do with the rescue operation, leaving the prospect of a diplomatic stand-off between the two EU allies.
Anelise Borges on the Aquarius
Euronews' Paris correspondent Anelise Borges was on board the Aquarius ship, operated by Doctors without Borders and SOS Mediterranee.
"This has become a high-level political negotiation," said Anelise. "The teams here onboard tell us that, for now, the situation onboard is under control. They can keep the people here, they have enough water and food for everybody, but this is a very delicate situation."
"There are more than 600 people onboard, they are tired, they want to get to safety, there is, of course, a lot of tension among them because they are not sure what is going to happen," she added.
Anelise praised the teams of staff on the vessel: "The teams here are very calm, they say the situation is under control and we continue heading north towards the east part of Malta and towards Italy."
At the time of writing, the Aquarius was stationary while its destination was thrashed out between the two countries.
'Enough supplies for two-three days'
MSF Sea Project Coordinator Aloys Vimard told Euronews that his team was "well prepared" and the boat contained enough food and water to house its passengers for between two and three days.
Nevertheless, he hoped the 629 people rescued can be transferred to a port of safety as soon as possible.
Who is on board?
The boat rescued the migrants in six different operations off Libyan shores on Saturday morning.
Hundreds of those were initially plucked from the sea by the Italian navy.
More than 600,000 migrants have reached Italy by boat from Africa during the past five years.
Anelise is sharing the Aquarius' location live via Twitter.