Eight former and serving officers have been called to give evidence at the hearings.
South Yorkshire Police and Crime Commissioner Alan Billings said the sum was "substantially less" than required.
The force, which has a legal duty to support its staff, said that as of January more than £12m had been paid out on legal representation.
The total spend had now risen to "considerably more than that", it added.
Dr Billings described the financial situation for the force as "grave" and said taxpayers in the region would "have to bear the additional cost".
He said it would ultimately impact on the police budget and the force's ability to provide policing and crime services.
Dr Billings said he would be calling an emergency meeting.
Ninety-six Liverpool fans died after crushing at the FA Cup semi final match between Liverpool and Nottingham Forest held at Hillsborough in 1989.
Fresh inquests into the deaths were ordered after the original accidental death verdicts were quashed in 2012 following an independent report.
The inquests are expected to continue for another year.
Dr Billings said: "The special grant application I submitted was for the full amount, which leaves us the shortfall to pay.
"My office now has to undertake a financial analysis to establish the size of the shortfall and its impact on our budget."
"While I recognise that the Home Secretary's decision was a matter of discretion, I am extremely disappointed by both the timing of the announcement and the amount of funding granted."
Dr Billings said he would submit another application to fund "the further legal costs we will incur during 2015/16".
A Home Office spokesman said: "On 27 March the home secretary agreed to provide a special grant to the South Yorkshire police and crime commissioner in relation to legal costs incurred by the inquests into the Hillsborough tragedy.
"The amount is up to £5.1m in 2013/14 and £5.6m in 2014/15."
Special grants can be provided to police forces facing unexpected or exceptional costs exceeding 1% of its annual budget.
South Yorkshire Police has been awarded £10.7m to cover legal fees incurred from the new Hillsborough inquests.