Since becoming unitary in 1998, the council says it has been severely under-funded by central government, which has underestimated its requirements.
The council has taken the unusual step of publishing detailed information about its financial predicament.
Leader Keith Lock (Lib Dem) has written to health secretary Alan Milburn and local government secretary Stephen Byers asking them to reassess its grant. They are seeking meetings with ministers and MPs. The council spends double the amount alloted in its standing spending assessment on community care.
In exchange for more cash, the council has offered to agree targets. A recent government review showed the shortfall was due to low SSA allocation.