There are also fears people are being turned away and many may not know how to access the grants.
Glasgow was allocated £2million this year from the fund set up by the Scottish Government.
But only 7% has been allocated since July.
In the first three months, only £131,161 was paid out, the second lowest in mainland Scotland with only East Renfrewshire allocating less cash at 6%.
Crisis grants replace crisis loans and are administered by the council not the Department of Work and Pensions as before.
MSPs heard how it is thought people are asking at the DWP for help only to be told the fund is no longer available.
Nicola Sturgeon, Deputy First Minister, was asked by MSPs on the Welfare Reform Committee about the concerns after she provided the figures detailing the spending in each council area. She said: "I know there are concerns and we are not seeing uptake at the level we want, but it is a new scheme, maybe we will see a spike in the winter months.
"I would be of the view that over the year we will have a fund that is under a lot of pressure.
"We don't want to see a postcode provision, it is a national fund administered locally."
The committee heard from officials from councils and citizens Advice Scotland on the Welfare fund.
Anil Gupta, chief officer of the Convention of Scottish Local Authorities (Cosla), said: " People who are applying go to DWP and they are told the fund has been abolished and are no signposted on to the successor fund.
"There is a logistical problem."
So far this year, 2185 crisis grants in the city have been awarded, which is 65% of all applications made.
Douglas Proctor, formerly of Cosla, agreed it was anticipated there would have been a greater demand for crisis grants, given the financial difficulties many people are experiencing.
He said: "It is a concern.
"The spend is still under what we where predicting. It is a new scheme and it needs time to bed in and develop.
"It is not the finished article and we need to talk to the Scottish Government about increasing uptake."
He said he believed that in the future the fund would be under greater pressure.
He added the problem was only an issue for the current year before changes are made in the allocation procedure.