The First Minister said she and Mr Cameron agreed the timescale at talks in Downing Street yesterday in order to ensure a deal is reached before May’s Holyrood elections.
SNP ministers have said Scotland may be hundreds of millions of pounds a year worse off if the deal is not right over the financial footing for the new powers and have threatened to block the package.
However, the UK government has insisted the arrangements would be fair to Scotland and the rest of the UK.
Ms Sturgeon and Mr Cameron also agreed to strengthen co-operation between the governments on security and intelligence matters, in order to protect the UK from the threat of terrorism.
The Joint Terrorism Analysis Centre in London will increase the number of reports it shares with the Scottish Government, and the First Minister will receive more frequent security briefings.
Ms Sturgeon, speaking after the talks in London with Mr Cameron, said that the SNP government would not sign up to “disadvantageous” arrangements, which she warned could lead to a reduction in Scotland’s budget.
However, Ms Sturgeon said that she and the Prime Minister would still seek to reach an agreement on the fiscal framework by the middle of February, when the Scottish Parliament goes into recess.
Ms Sturgeon said: “We want a deal and if we’re going to get a deal before the Scottish elections, we need to reach that by the middle of February.
“I was very firm that that has to be a fair deal for Scotland, and we will continue in good faith to try to reach that.”
Ms Sturgeon added: “The proposal that has been on the table would lead to a systemic and inherent reduction in the Scottish budget, so regardless of the wisdom of the decisions that this or future Scottish Governments took, there would be a downward pressure on the Scottish budget.
“Now, I can’t sign up to that. I can’t in all good conscience agree to something that would be so disadvantageous to Scotland and to Scotland’s budget.”
Scotland Office minister Lord Dunlop said the Prime Minister was committed to “a good deal both for Scotland and for the rest of the UK”.