(Updates with JBS saying it had no comment)
By Brad Brooks
SAO PAULO, May 28 (Reuters) - Brazilian federal prosecutors on Sunday made a new offer to JBS SA's controlling shareholder, J&F Investimentos, that it pay a 10.99 billion real ($3.37 billion) fine for its role in massive corruption scandals.
The new offer is down slightly from the previous proposal by prosecutors that J&F pay 11.2 billion reais. The company rejected that and counter-offered, saying it would pay 4 billion reais. Prosecutors rejected that, as well as the company's next offer that it would pay 8 billion reais.
J&F told Reuters in an emailed statement that it had no comment on the matter.
Negotiations on the leniency deal fine follow bombshell state's witness testimony from J&F's owners Joesley and Wesley Batista that they spent 600 million reais to bribe nearly 1,900 politicians in recent years.
Joesley Batista is at the center of a corruption investigation into President Michel Temer. Batista secretly recorded a conversation with Temer in March in which the president seemed to condone bribing a potential witness in the corruption case.
The top court authorized a corruption investigation into Temer based on that tape and testimony that the Batista brothers and five other executives from their company gave regarding the president and several other powerful politicians.
Investors are watching the plea negotiations. JBS shares have slid more than 20 percent since mid-May in extremely turbulent trading because of concern that blowback from the scandal could limit its funding options.
The bribery scandal is one of several legal hazards facing the group, which mushroomed from a regional Brazilian slaughterhouse to the world's largest protein processor with the help of some 8 billion reais in government support.
Brazilian rules for corporate leniency deals call for fines of somewhere between 0.1 percent and 20 percent of annual sales.
Prosecutors said in their written statement on Sunday that the proposed 10.99 billion real fine was equivalent to 6 percent of J&F's group 2016 revenue.
Authorities also for the first time detailed who would receive the fine from J&F. Brazil's BNDES state development bank, along with the pension funds Funcef and Petros would each receive 25 percent of any fine.
The remaining 25 percent would go to the federal government, which would get 12.5 percent, along with the FGTS worker severance fund and the Caixa Economic Federal bank, which would each receive 6.25 percent.
$1 = 3.2598 reais Reporting by Brad Brooks and Tatiana Bautzer in Sao Paulo, and Lisandra Paraguassu in Brasilia; Editing by Dave Gregorio and Mary Milliken