The UN chief has called for restraint as some 700 activists from around world vow to deliver 10,000 tonnes of humanitarian aid to break the blockade of Gaza.
Israel has cautioned that the Freedom Flotilla would be stopped, if necessary by force.
The nine-ship flotilla is by far the largest fleet of aid to try to reach the coastal Palestinian territory since Israel imposed its siege on it in 2007.
"We strongly urge that all involved act with a sense of care and responsibility and work for a satisfactory resolution," a spokesman for Ban Ki-moon said on Thursday.
He restated UN opposition to the siege of Gaza and the lack of material to meet "basic needs, begin reconstruction, and revive economic life".
After the Israeli army announced a detention centre at Ashdod port for holding the activists, Greta Berlin, one of the flotilla organisers, said: "We have the right to sail from international waters into the waters of Gaza.
"The only illegal presence in the area is Israel."
Berlin said the Freedom Flotilla was on schedule to arrive in the Gaza Strip on Saturday with more than 10,000 tonnes of supplies, including water-filtration units and pre-fabricated homes.
Israel and Egypt have sealed Gaza off to all but very limited humanitarian aid since Hamas, the Palestinian political faction, took control of the territory in June 2007.
Israel says the Gaza blockade aims to prevent Hamas from acquiring weapons or materials that could be used for military purposes.
For the majority of Gaza's population of 1.5 million people, the result has been impoverished living conditions.
Catherine Ashton, the EU foreign policy chief, issued a statement on Friday, calling for an immediate end to Israel's blockade on Gaza.
"We would like to reiterate the EU's call for an immediate, sustained and unconditional opening of crossings for the flow of humanitarian aid, commercial goods and persons to and from Gaza," she said.
"The continued policy of closure is unacceptable and politically counterproductive.
"The EU remains gravely concerned by the humanitarian situation in Gaza."
Israel's foreign ministry said it had given warnings to the ambassadors of Cyprus, Greece, Ireland, Sweden and Turkey - from where the ships set sail - that it had "issued warrants that prohibit the entrance of the vessels to Gaza".
The flotilla "is about to break international law", Yossi Gal, the ministry's director general, said.
Gal said that the flotilla was "an absolute provocation" and a "cheap political stunt", as there was no shortage of humanitarian aid in Gaza.
Al Jazeera's Sherine Tadros attended an Israeli army news conference on Wednesday, where journalists were told that there was no humanitarian crisis in Gaza.
She said that the information to back up this claim was both incomplete and out of context: "This tells me what Gaza is getting in terms of supplies but does not compare this to how much Gaza needs to survive."
Israel has vowed to divert the ships to the southern Israeli port of Ashdod.
It has said that Israelis on board would be arrested, Palestinians would be questioned by the Israeli secret service, and foreign nationals deported.
Part of the port has been cordoned off and prepared to deal with the activists, and large tents set up for immigration booths and areas for people to be searched.
Gal suggested the organisers should voluntarily head to Ashdod to unload the supplies so Israel or humanitarian agencies can deliver them to Gaza overland, but the flotilla organisers rejected the offer.
Hanin Zuabi, a member of the Israeli parliament who is on board the flotilla, told Al Jazeera that the activists intend to reach Gaza regardless of plans to stop them.
"If the Israelis try to stop us, this will be a huge diplomatic and political crises for them," Zuabi said.
"We have 50 states participating in this and are sending a very clear message to Israel - the international community is not accepting the siege on Gaza."
Peace laureates aboard
Al Jazeera's Jamal Elshayyal, on board the flotilla, said the activists travelling in the convoy included European parliamentarians, former US diplomats and Nobel peace laureates.
Berlin, the flotilla organiser, said: "This mission is not about delivering humanitarian supplies, it's about breaking Israel's siege on 1.5 million Palestinians."
Fintan Lane, an Irish activist, said that they were determined to break Israel's blockade and will not be intimidated.
"The people of Gaza have a right to access to the outside world and the right to determine their own future," Lane said.
Huwaida Arraf, one of the organisers from the Free Gaza Movement, said: "Israel should not be under any illusion whatsoever that their threats or intimidation will stop us or even that their violence against us will stop us."
Some Israeli officials see the situation as potentially disastrous in terms of public relations.
"We can't win on this one in terms of PR," Yigal Palmor, a foreign ministry spokesman, said.
"If we let them throw egg at us, we appear stupid with egg on our face. If we try to prevent them by force, we appear as brutes."
Hamas officials say that Israel's threats to intercept the flotilla amount to "Zionist piracy".
"The occupation's threat to prevent the Freedom Flotilla from arriving in the besieged Gaza Strip is Zionist piracy and a violation of international law," Ismail Radwan, a senior Hamas leader, said.
"The occupation is concerned about these ships... because they grant legitimacy to engagement with the Palestinian government and confirm that the attempts to isolate Hamas have failed."