Earlier this week the Evening Times reported that Cosla officials told MSPs people in need were not being directed to the new fund by the Department of Work and Pensions staff.
MSPs also heard how Glasgow had the lowest rate of uptake for crisis grants with concerns the fund was underspent.
However, Glasgow City Council said the current position shows demand is rising with every month and the fund will not be underspent.
MSPs raised official figures that showed only 7% of the budget for crisis grants had been allocated in the city.
However, the council now says the most up-to-date figures shows more than 25% of the total welfare fund, which also includes Community Care Grants, has been spent so far.
A spokesman for the council said: "Glasgow looks at the welfare fund as one budget and total spending now stands at over £2m.
"The majority of the claims we have received have been for community care grants, which assist with sustainable living.
"Like other local authorities, Glasgow experienced a slow uptake from the fund to begin with.
"But demand is now rising month on month and we do not anticipate the fund will be ultimately underspent."
The city received £7.7m for the welfare fund and has spent more than £2m.
In total, up to the end of September 6600 crisis grants have been awarded, which amounts to £240,000.
The figure is almost double the £131,000 that was given to MSPs in a report.
Meanwhile the claim by Cosla, chief officer, Anil Gupta that DWP staff were not pointing claimants to the new fund was branded "nonsense".
A spokeswoman for the DWP said: "We reformed the Social Fund because it had become complex and poorly targeted and so replaced it with funding for new local provision to ensure that money is targeted at those most in need.
"We transferred the current annual funding for crisis loans for living expenses and for community care grants to the Scottish Government, who received the full programme budget for 2013/14 and 2014/15 in addition to administration and start-up costs, amounting to almost £24m.
"It's nonsense to suggest our staff don't signpost claimants to this provision from their local authority."