French riot police have arrived at a Calais port terminal to try to move dozens of Syrian refugees who have been blockading a walkway to the port building for three days, demanding access to British officials in order to seek asylum in the UK.

The approximately 60 Syrians include seven minors and a few women – one pregnant – with the rest being men, at least 20 of whom are on hunger strike.

Many had taken months to travel through Turkey and the Balkans in an attempt to reach Britain, but found themselves stuck in Calais in makeshift squats in abandoned buildings, or sleeping rough after being evicted by French authorities.

Angered at their poor conditions, they picketed the terminal, where ships leave for British ports, with cardboard signs reading "Take us to the UK" and "We want to talk to David Cameron".

Officers stand by as two refugees sit on the roof of a ferry terminal, threatening to jump if police try to move them. Photograph: Philippe Huguen/AFP/Getty Images

When riot police arrived early on Friday, two of the Syrian men climbed on to a roof and threatened to jump as police tried to negotiate with them to come down. The local prefect offered the Syrians the right to apply for asylum in France, but they continued their protest.

Philippe Wannesson, of the French association La Marmite aux Idées, which helps refugees in Calais, was at the terminal. "Their aim is to get to the UK and they want to speak to British representatives," he said. "More and more Syrians are arriving in Calais wanting to reach the UK. A year ago we saw perhaps only two, three or four. Since March or April it has been rising to 20, 30, 40 and steadily growing."

Wannesson said a delegation of Syrian refugees and charity workers spoke to UK Border Force representatives in Calais in the afternoon but the Syrians remained at the ferry terminal and felt "let down". They were meeting to vote on continuing the protest.

One refugee, a 24-year-old English literature graduate from Aleppo, who did not want to be named, told the Guardian by phone from the terminal: "I arrived two days ago and followed people here. It took me four months to make my way here from Turkey.

"The conditions here are very bad. People have had a lot of difficulty with the French police, they have been arrested or followed, they have had to sleep rough in abandoned buildings or on the street, sometimes for months.

"We just want to appeal for asylum in the UK. We don't hold any hope from the French government. We hope that the British government will show some of the humanity it is famous for. People are desperate. The two people on the roof threatening to jump are proof of that – one has lost two brothers, he has nothing left to lose. I'm so worried about my family back in Aleppo but they wanted me to go."

In a statement released through the Calais Migrant Solidarity group, the demonstrators said: "We demand one person from the UK Home Office comes here to speak with us, and to see our situation. We have the right to claim asylum in England, but how do we get there? There is not a legal way to cross."

UK authorities said the protest was a matter for the French.

A UK Border Force spokesperson said: "The responsibility for legal and illegal migration in France is the responsibility of the French authorities.

"We work very closely on matters of border security with the French authorities to maintain the integrity of the controls."

The United Nations refugee agency has said 17 countries, including France and the US, have agreed to receive quotas of refugees fleeing the conflict in Syria. They have offered to take in a total of 10,000 people. About 2 million Syrian refugees have fled to neighbouring countries, such as Turkey, Lebanon and Jordan.

At the G20 meeting in Saint Petersburg last month, David Cameron announced an additional £52m in UK aid for those affected by the civil war in Syria, saying: "This is a moral imperative. This is the big refugee crisis of our time."