Immigration minister Caroline Nokes has agreed to allow Glasgow City Council to individually assess all asylum seekers embroiled in an eviction row.
Public service provider Serco has said it is providing housing for 330 people “at its own expense”.
Ms Nokes met the council leader on Tuesday to discuss the eviction process for those not given refugee status.
Serco has temporarily suspended plans to change the locks on properties after legal action and protests.
Housing charity Shelter Scotland has filed papers at Glasgow Sheriff Court to prevent two tenants being issued with so-called lock-change orders.
The issue is also due to be raised in the Court of Session in Edinburgh by Govan Law Centre, which wants to establish the legal principles involved.
Following the “constructive” meeting with the UK immigration minister, council leader Susan Aitken said: “There is currently little clarity about the actual status of the 330 people affected, therefore it is essential that Glasgow City Council has the time and opportunity to carry out individual assessments for all of them and ensure that everyone gets the right support and outcome for their circumstances.
“The minister agreed this was necessary and her officers have committed to engaging constructively with us to ensure this is put in place.”
She added: “Ms Nokes and her officials have been left in no doubt that lock changes and evictions are not good for anyone, and do not have support within the city.
“Whilst she was unable to give a guarantee that they would not continue, we have made significant progress.
“However, we will continue to join with partners across the city to campaign for this policy to be changed.”
The minister was due to meet Serco later, and hold talks with the Scottish government on Wednesday.
Serco is contracted to provide housing while applications are considered by the Home Office.
Last month, it said it planned to evict those tenants who had exhausted the asylum process.
Ms Aitken wrote to Home Secretary Sajid Javid warning that it risked a “humanitarian crisis”, leaving those affected out on the streets.
Serco denied the claim and said it currently pays accommodation for 330 asylum seekers who no longer receive Home Office support.
About a third have been granted leave to remain while the remainder have been refused refugee status.
Serco’s eviction plans affects the latter group, which the firm says has no right to remain in the UK.
Scottish Labour leader Richard Leonard has written to Ruth Davidson, urging her to “break her silence” on the issue and put pressure on the government to halt the evictions.
In response, the Scottish Conservative leader said she met Serco chief executive Rupert Soames last week to discuss the eviction proposals.
She endorsed statements from leader of the Conservative group on Glasgow City Council, David Meikle, calling for clarity and a solution to the “potential crisis”, and from Tory MSP Adam Tomkins.
Mr Tomkins said: “If people whose applications have been rejected are housed indefinitely, that is manifestly unfair to others with genuine asylum claims.”
Ahead of the meeting with Ms Nokes on Wednesday, Aileen Campbell, cabinet secretary for communities and local government, said: “An asylum system that sees people end up destitute and homeless in the country where they sought refuge lacks humanity, and the UK government must be open and accountable for its decisions and policy, which remain wholly reserved.
“I want to see changes that will deliver a fair, effective and compassionate system that treats everyone seeking asylum with respect while their legal rights are fully met.”
She added: “The current situation in Glasgow is a human rights issue at which people across Scotland have quite rightly shown outrage.
“As such, I will press the UK government to fix their failing system by finding a long-term solution and ensure the current situation cannot happen again.
“This means the Home Office must ensure equity of funding for Glasgow City Council which is a dispersal area. It is unacceptable to deny funding that is available to English councils to a council which has welcomed asylum seekers and given them a home.”
The Home Office earlier said asylum seekers “who would otherwise be destitute” are provided with free accommodation while their applications are considered.
It added: “While an asylum claim or an appeal is outstanding, we would not be seeking removal.
“If the courts have decided that someone has no right to remain in the United Kingdom it is right that they should leave the UK.
“However, if an asylum claim has failed, we will still provide accommodation for those who would otherwise be destitute and who are temporarily unable to leave the UK because of a practical or legal obstacle.”