Greek authorities were facing growing pressure on Friday over claims they could have acted earlier to help migrants on a ship that sank this week, killing at least 78 people and leaving hundreds missing.

As hopes faded for further survivors from the overcrowded fishing vessel that capsized off Greece on Wednesday, a group supporting migrants attacked the authorities’ response to the boat and their account of events leading up to the disaster.

At least 100 women and children in the vessel’s hold are believed to have died, while estimates of the total dead range as high as 600. Thousands of people protested across Greece on Thursday against the authorities’ handling of the accident and the EU’s tough policies on migration.

Alarm Phone, a network of activists that helps migrants in trouble to call for help, said Greek authorities “had been alerted many hours before the vessel capsized and had been informed by different sources that this was a boat in distress”.

“European authorities could have sent out adequate rescue resources without delay. They failed to do so because their desire to prevent arrivals was stronger than the need to rescue hundreds of lives,” the group said.

Nikos Spanos, a retired naval officer from the coast guard, told national broadcaster ERT that Greece should have intervened.

“We don’t ask the crew in a boat that is in danger if they need help. They needed help [ . . . ] and we needed to do the right thing,” Spanos said.

Greek officials said offers of assistance were repeatedly rejected by those on board. “They categorically refused any help,” said Nikos Alexiou, a coast guard spokesman. He said a forced intervention of a vessel full of people could have led to its sinking.

The country’s minister for civil protection, Evangelos Tournas, said the vessel had been in international waters and Greece was unable to intervene when the crew refused help.

Alarm Phone and the coast guard both released their accounts of contact with the vessel ahead of the disaster, with the NGO’s account suggesting people on board the ship were making distress calls several hours before it sank.

The Greek coast guard said it was informed of the boat’s presence by Italian authorities on Tuesday morning. An aircraft from Frontex, the European border and coast guard agency, had detected the boat shortly before 10am local time.

On Tuesday afternoon, the Greek coast guard contacted someone on the boat who said those on board needed food and water but wanted to continue their journey to Italy.

However, Alarm Phone said it was contacted by people on the vessel seeking help shortly after 3pm that day. They said they “cannot survive the night”, Alarm Phone said.

At 5.20pm, Alarm Phone contacted the fishing vessel, where people on board said that “the captain [had] left on a small boat”. The Greek coast guard ordered two container ships near the vessel to approach it. At 6pm one of them provided food and water to the boat.

At 10.40 on Tuesday evening, a Greek coast guard vessel approached the fishing boat and reported it was travelling at a stable speed and direction.

But at 1.40am on Wednesday, the coast guard vessel reported that the migrant boat had stopped moving, and at 2.04, the fishing vessel banked steeply to the left and then the right, resulting in its overturning.

The coast guard also dismissed media reports citing survivors who said a rope attached to the vessel by the coast guard had contributed to its sinking. It said the rope had been quickly untied by passengers.

A vast search-and-rescue operation recovered 78 bodies and 104 survivors, but more than 72 hours after the accident, hopes were fading for any more victims to be found alive.

Estimates of the number on board have varied from 400 to 700 people, with survivors estimating that more than 100 children and women were in the ship’s hold, making the accident one of the deadliest in the Mediterranean.

The rescued passengers, all males including eight minors, were from Afghanistan, Egypt, Pakistan, the Palestinian territories and Syria, the coastguard said.

Greek authorities arrested nine suspects, all Egyptian men, on Thursday on suspicion of people smuggling and organising the deadly voyage. The ship began its journey in Egypt and stopped in Tobruk, Libya, where it picked up the migrants heading to Italy.