The offices in Easterhouse, Pollok, Castlemilk, Parkhead and Bridgeton will shut on March 31 and mean the loss of 42 jobs.
The move comes after the city council had invited organisations to bid for providing advice services to local people.
The tender was split into four and a consortium of the eight city CAB offices, along with other groups, were awarded two.
The other two went to the Glasgow Advice Agency consortium of the Greater Easterhouse Money Advice Project, Money Matters Advice Centre, Govan Law Centre and the Drumchapel Law And Money Advice Centre.
Over the past few weeks, the five bureaux have been in unsuccessful discussions with the consortium to try to arrange sub-contracts to secure their future.
A spokesman said: "The bureaux now have to close when their funding runs out at the end of March."
However, Glasgow City Council leader Gordon Matheson said there was no need to close the offices.
He said: "The council has guaranteed continued funding for these centres and there is no need for a single one to close.
"But I will ensure people living in those parts of our city will have access to those vital debt and money advice services."
The CAB spokesman said it had no alternative but to shut the offices.
He added: "We are keen to talk to anyone who can help solve this problem, but the council's current proposal does not do that."