Universal Credit, the UK Government's flagship welfare reform policy, is intended to be introduced in the city in October.
But Glasgow housing bosses told UK Work And Pensions Secretary Iain Duncan Smith that is too soon.
The overall reforms are expected to save almost £30billion for the Treasury through cutting costs and reducing fraud.
But some of the expected savings from the introduction of Universal Credit, as well as the cuts to benefits, will come because administration depends on people having internet access to make applications and bank accounts to receive payments.
Glasgow Housing Association chairman Gordon Sloan and chief executive Martin Armstrong met Mr Duncan Smith to make him aware of the situation and asked him to make Glasgow a case for special consideration.
They expressed concerns there are still too many people in the city with no bank account or proper experience of managing money over time through an account. They also told him too few people have internet access or computer knowledge to negotiate the application process online.
As well as the practical issues, they warned of the severely damaging impact the reforms would have on tenants on low incomes and on housing providers if, as expected, it leads to greater financial difficulties.
The GHA bosses told Mr Duncan Smith that as many as 30,000 tenants in Glasgow do not have a bank account and deprived communities have insufficient provision to cope with the increase in demand.
Mr Duncan Smith was also told total broadband take-up in Glasgow was one of the lowest in the UK, with just over half of homes online – well below the average south of the border, where almost three quarters of homes have internet access.
Mr Sloan said: "There are two inescapable facts: One, these cruel and unfair welfare changes are coming and, two they will have a devastating impact on families and communities across Glasgow.
"The plans to igmplement Universal Credit, in particular, will have a challenging and far-reaching affect on Glasgow and especially 28,000 of our tenants.
"We made it clear to Mr Duncan Smith that Glasgow would be harder hit than any other part of the UK and it should be considered as a special case on the basis of the deprivation levels that still exist in many parts of the city.
"Iain Duncan Smith has visited Glasgow several times over the years, including a much-publicised visit to Easterhouse in 2002.
"He has seen at first hand the massive challenges many people and communities face in dealing with his reforms. There is do doubt we need more time to help our tenants to cope with these changes."
In our series of articles on welfare reform, the Evening Times has been highlighting the proposals and the potential for even greater poverty and reporting where people can go for help.
Paying benefits in one monthly sum and making tenants responsible for paying their rent instead of direct to the landlord is expected to push many further into financial difficulties because they are not used to managing money.
Mr Sloan added: "At GHA, we know from talking to tenants that many fear receiving their benefits directly in one sum every month. They worry they will not be able to budget and manage their financial affairs and will run out of money before the end of the month."
Universal Credit will replace several benefits, including child and working tax credits, Job Seeker's Allowance and Housing Benefit with a one-off monthly payment.
The Government says it will reduce fraud, smooth the transition between work and benefits and make work pay.
Housing groups and advice centres, however, have warned they will be struggling to cope with the demand once tenants and benefit claimants are made fully aware of the implications on their income.
More than one million families are expected to be affected across Scotland, with Glasgow Housing Association estimating the reforms will hit 28,000 of its tenants.
Mr Armstrong, chief executive of GHA parent company Wheatley Housing Group, said: "Social landlords across Glasgow and beyond are doing their bit with key partners, such as the city council, to prepare tenants for the changes.
"We are doing everything we can to help tenants cope with the impact of welfare reform.
"But, given the scale of the challenge and the number of people affected, we need the Government to recognise Glasgow is a special case and needs more time to develop the infrastructure to support the people affected."
GHA has launched a free helpline for its tenants with landline phones to get advice. The number is 0800 5973777.
Those with mobiles should call 0141 444 0110 for a cheaper rate.
Staff are talking with all tenants, especially those who will be directly affected by the impact of the under-occupancy rule – the so-called 'bedroom tax' – and explaining the options there are to lessen the impact.