David Crompton was asked to resign from his position as chief constable of the force by Police and Crime Commissioner Dr Alan Billings following the verdict of the Hillsborough inquests, which found 96 Liverpool FC fans were unlawfully killed on April 15, 1989.
Mr Crompton is seeking a judicial review at the High Court, which is due to be heard on March 28 and 29.
Five Hillsborough relatives, Dorothy Griffiths, Barry Devonside, Becky Shah, Wendy Hamilton and Charlotte Hennessy, asked two judges in London on Wednesday to allow them to make written and oral submissions at the pending hearing, but were refused permission.
Lawyer for the families Elkan Abrahamson said the five relatives had been ordered to pay Mr Crompton's legal costs, which had been estimated as £28,000, although the figure was due to be negotiated.
A Justgiving page set up by friend of the families Diane Lynn has already raised more than £4,500 towards the costs.
Any extra money raised will be donated to Alder Hey Children's Hospital.
Mr Abrahamson said: "It's great people are prepared to help the families.
"Mr Crompton is entitled to ask for his costs, although none of the other three parties involved in the case did.
"That amount of money is a lot for anyone to find."
Donations can be made here.