The umbrella body for CABs in Scotland said this week that the bureaux would close at the end of the month unless emergency funding was found.
The announcement came after the council awarded contracts to provide advice services to a consortium called the Glasgow Advice Agency. The council hoped the GAA would subcontract work to the bureaux.
When news of the impending closures broke, the GAA wrote to the five organisations offering three-year funding at a lower level than they currently receive.
However, the council has pledged to provide transitional funding of £250,000 a year over the next three to top up the value of the contracts to existing levels.
A spokesman for Citizens Advice Scotland said the bureaux were looking carefully at this new proposal, but some concerns remained.
He said: "They will not be responding formally to the offer until they have had the chance to take legal advice and consider the proposals in detail.
"The fact we now have a formal legal document is a real step forward, as is the fact that they are now talking about a three-year proposal, not just one.
"However, some concerns remain. This is not yet an actual contract, the funding being offered still represents a significant year-on-year cut and the terms and conditions need to be clarified."
A council spokesman said he was unable to say why the transitional funding was being provided in addition to the funding offered to the CABs by GAA.
He said: "It’s absolutely vital that the people of Glasgow have access to the money advice they need, especially during the current financial environment.
"That’s why we’ve made sure that there will be absolutely no need for any CAB to close."