The latest demand comes from local and southern state governments, including Louisiana and Mississippi, hit hard by the pollution that followed the blowout on the BP-operated Macondo well in April 2010.
BP described the methodology for calculating the claims as "seriously flawed" and the oil company said it was confident it would not have to make additional financial provisions.
BP refused to say what provisions it has already made for the $34bn claims, but admits to having taken a further $4.1bn "charge" in the final three months of 2012, bringing the total amount set aside so far to $42.2bn.
The oil company has settled criminal charges with the US department of justice, but will suffer further financial hits as it prepares for a final civil trial scheduled to start on 26 February.
Worst case scenarios could mean a further $20bn of liabilities in the event of the company being found "grossly negligent", but BP expects any fine to be very much smaller and is still desperately working on an out-of-court settlement to avoid the trial.
The latest numbers resulting from the Deepwater disaster were released by BP on Tuesday morning as part of its fourth-quarter and full-year financial results, which showed the company had already sold $37.8bn of assets even before its sale of its half share in the Russian joint-venture TNK-BP to Rosneft.
That disposal has not been completed yet, but BP's overall production – excluding TNK-BP – was down 7% in the final three months to 2.29m barrels of oil equivalents a day.
The output figures were damaged by the sale of assets to pay for the Macondo blowout, and they helped drive down quarterly underlying profits by 20% in the final period, to $4bn from $5bn.
The $4.1bn Deepwater charge, however, was considerably lower than analysts had forecast, and BP chief executive Bob Dudley gave an upbeat assessment of what the company had achieved.
"We have moved past many milestones in 2012, repositioning BP through divestments and bringing on new projects," he said. "This lays a solid foundation for growth into the long term."
BP said it was already in talks with Rosneft, in which it will take a 20% stake, about joint venture operations and said it could bring new technology and good governance to the Moscow-based group.
Dudley would not rule out moving into the Russian Arctic with Rosneft given the scale of potential oil and gas reserves there but insisted it would treat the pristine environment "with great sensitivity and care".
Dudley also said he did "not rule out getting involved in UK shale" and supported the British government strategy of looking at ways to encourage gas drilling for these more controversial resources.
The bill so far for the Deepwater Horizon disaster
1. Beach and other cleanup costs – $14bn
2. Early claims settled – $9bn
3. Early restoration projects – $1bn
Plus major legal settlements:
4. Criminal charges with the department of justice – $4bn
5. Individual and small businesses claims through a plaintiff steering committee – $7.8bn
6. Securities and Exchange Commission claims – $525m
7. Claims from local and state governments – $34bn
8. Worst case scenario damages under a Clean Water Act – $21bn
9. Natural resource damages – (unquantified)
BP has won back $5bn from well equity partners Anadarko and Mitsui.
No settlement yet with rig operator Transocean or contractor Halliburton.
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image: © Deepwater Horizon Response