The Senate has passed a motion calling on the government to recognise the increased risk it has placed on two gay Saudi journalists by keeping them in detention after they claimed asylum last month.

Guardian Australia can reveal that multiple news organisations around the world have called for the release of the men, warning the Australian government they are watching the case closely.

The motion, brought by Greens senator Janet Rice, also urged the government to expedite the refugee assessment process for the couple and assure their safety in the interim.

Labor, Centre Alliance and independent senator Jacqui Lambie all pledged their support for the motion, which passed on Wednesday afternoon.

Last month Guardian Australia revealed the two men were detained at an Australian airport after passing through immigration on valid tourist visas. They have remained in detention or under guard in hospital ever since, and have alleged threats of violence against them as well as fear they could be targeted by Saudi representatives.

Sultan* is a former ministry of media employee and a fixer who had worked with multiple international media organisations, and Nassar* is a cameraman.

They fled their home country after Sultan was interrogated by Saudi authorities, who ordered him to stop working with foreign media and made veiled threats to out his and his partner’s relationship.

Homosexuality is illegal in Saudi Arabia and punishable by death.

Sultan told Guardian Australia last month their arrest and “arbitrary” detention in Australia was comparable to threats back home.

“We’ve been threatened with it in Saudi but it never actually happened until we came here,” he said.

Rice welcomed the Senate’s move.

“When these journalists liken their treatment in Saudi to the violence they have experienced here in Australia, it should be a wake-up call,” she told Guardian Australia.

“The government should act swiftly to reflect the view of the Senate on this matter and uphold the values of fairness and diversity that our country says we stand for.”

“This motion is an important step to increased public awareness of how dangerous Australian immigration detention centres are,” said the men’s lawyer, Alison Battisson.

“I am aware of serious assaults, requiring hospitalisation, taking place on a weekly basis … When assaults do occur, it appears very difficult for the Australian federal police to access detention to investigate.”

Several international media organisations have also written to Australia’s immigration minister, David Coleman, in support of the men.

“This is unjust treatment of two journalists who travelled to Australia seeking protection from the persecution they faced as gay men in Saudi Arabia,” wrote one letter from Mark MacKinnon, of Canada’s Globe and Mail.

“I hope you can intervene to ensure they are released from detention – they have valid visas to visit Australia – so that they can pursue their asylum claim,” he said.”

“Please trust that I and my colleagues around the world who know and have worked with [Sultan] will be watching the outcome of this process with very keen interest.”

Other letters from some of the world’s most prestigious newsrooms, including the BBC and ITN, urged Coleman to grant the pair asylum, saying the pair would “fit in very well in Australia”.

Michael Garrod, the founder of freelancer database, World Fixer, said Sultan was driven to help foreign media “in the spirit of progress, driven and inspired by the initiatives of the Crown Prince”, and said if the couple was returned to Saudi Arabia they faced persecution because of both their sexuality and their profession.

“I sincerely hope that your department recognises their considerable contribution to our global understanding of their country and stands by Australia’s promise to support LGBT rights globally,” he said.

Peter Greste, journalist and spokesman for the Alliance for Journalists’ Freedom, said: “If Australia wants to be seen as a campaigner of press freedom around the world, and wants to be respected and taken seriously, the government will act swiftly, and with understanding and compassion, in relation to these two journalists.”