Rahman Shah, 32, and Mirwais Ahmadzai, 27, said they would remain outside the office complex in Brand Street for “as long as they physically could”.
Both men are Afghans who claim to have fled the on-going war in their homeland, only to see their asylum claims rejected after arriving in the UK.
Mr Shah is among those to have received a letter from Serco informing him to vacate his temporary accommodation in Glasgow or face eviction.
The private firm last week began issuing seven-day notices to the first of 300 cases in Glasgow informing them their locks are to be changed.
The company says it has been providing accommodation at its own expense after the Home Office withdrew funding for those refused leave to remain in the UK.
Serco, which is currently accommodating 5,000 asylum seekers in Glasgow on behalf of the Home Office as part of its Compass contract, said it had been working on the eviction process with the city council for the past four months.
Mr Shah said he fled an refugee camp in Pakistan after authorities there told him to return to Afghanistan. He arrived in the UK last year, and claims the UK Border Force has since informed him he should claim residency in Pakistan.
Mr Ahmadzai arrived in Scotland aged 15 and has spent 12 years living in various temporary addresses. His MP, Labour’s Paul Sweeney, has raised his case with the Home Office.
Mohammad Asif, chairman of the Scottish Afghan Society, said he was working with both Mr Sahah and Mr Ahmadzai to resolve their issues.
“The issue is as much with the Home Office as it is with Serco,” he told The Scotsman. “These men have fled a war-torn country and yet the UK Government is asking them to return.
“The Foreign Office advises British nationals not to travel to Kabul, yet this is where they want Afghans to return to.”
Mr Sweeney, the Labour MP for Glasgow North East, said “direct action” was necessary because the proposal to evict residents had been announced during the parliamentary recess - meaning the normal political channels could not be relied on.
The Home Office said it was not “seeking removal” of anyone appealing their decision on leave to remain, but Mr Sweeney said a constituent who was appealing had been served with an eviction notice.
“He’s been in this country for the last 12 years,” he said. “He came from his country [from Afghanistan] as an unaccompanied minor aged 15.
“Unfortunately the Home Office rejected his application in May. He’s going to appeal it, his lawyer has written [confirming the appeal] earlier this month... he’s been served with an eviction notice so clearly something’s gone badly wrong there.
“I’ve got reports here from a consultant psychologist saying the guy has actually attempted suicide in the past, he’s got mental health problems. So he’s also very vulnerable.”
The Home Office said it would not comment on individual cases, but a spokeswoman said: “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.”
A spokesman for Serco said the firm had only served eviction notice on “six male adults, all of whom our information from the Home Office - who provide the information, we’re not involved in that process - is that their claim for asylum has failed”.
He added: “Without knowing the details of the individual case that’s being talked about, I honestly can’t comment on it.”
Additional reporting: Conrad Landin