Although the navy did not attempt to intercept the flotilla's vessels, organisers of the attempted siege break said they diverted their ships and slowed down to avoid a confrontation during the night.
They also issued all passengers life jackets and asked them to remain below deck.
Hundreds of pro-Palestinian activists, including a Nobel laureate and several European legislators, are with the flotilla, aiming to reach Gaza in defiance of an Israeli embargo.
But Israel has said it will not allow the flotilla to reach the Gaza Strip and vowed to stop the six ships from reaching the coastal Palestinian territory.
The flotilla had set sail from a port in Cyprus on Sunday and aimed to reach Gaza by Monday morning.
Two other ships were damaged over the weekend, and remained in port in Cyprus.
The organisers of the fleet, dubbed the Freedom Flotilla, said they might launch a second smaller convoy of boats on Tuesday, which would include the two damaged ships, plus a third that had yet to arrive.
The flotilla was originally made up of nine ships - from Turkey, the UK, Ireland, Greece, Kuwait and Algeria - carrying around 10,000 tonnes of aid, including cement, water purification systems and wheelchairs.
It was initially expected that the flotilla would set sail on Saturday, but it was delayed over the weekend due to mechanical problems and was forced to anchor off the coast of Cyprus.
Hamas welcoming committee
Hamas, the de facto rulers of the Gaza Strip, have said that the flotilla was about to make history, sending "a strong message that the blockade imposed on the Gaza Strip ... will be broken".
Nicole Johnston, Al Jazeera's correspondent in Gaza, said there was a great deal of excitement there about the flotilla.
"There have been preparations going on at the port, drills, a Hamas welcoming committee ... but nobody knows if they will actually see this flotilla," she said.
"But one thing that the people of Gaza do appreciate is the international solidarity that they are feeling.
"It reminds people that they haven't been forgotten by the international community."
Israel said the boats were embarking on "an act of provocation" against the Israeli military, rather than providing aid, and that it had issued warrants to prohibit their entrance to Gaza.
It asserted that the flotilla would be breaking international law by landing in Gaza, a claim the organisers rejected.