The League One club is in the midst of a boardroom and financial crisis that has seen owner Dave Allen resign as chairman and four further shareholders resign as directors within a week.
Carson is quoted by Sky Sports News as saying: "We are in a situation where we have got to find funding and quickly. All week we have been looking for somebody to invest in the club, and nobody has come forward. Nobody has found anybody that wants to buy Chesterfield.
"Between now and Christmas, we've got to find a shortfall of half a million pounds. We'll keep looking for money and I'll keep talking to Dave Allen about this. The last thing he wants is to see the club go into administration, and that's the last thing we want as well.
"We haven't got a lot of time. We've got to find some money quickly. There're bills and wages to be paid. I can't put a time frame on it, all I can say is that we will continue to work extremely hard on this."
The Spireites lost 2-1 at Fleetwood Town on Saturday, leaving them firmly rooted to the bottom of the third tier.