On a final push to rally support in Europe before Thursday’s summit in Brussels, which could sway Britons voting in a referendum to stay or leave the EU, Cameron met Hollande in Paris on Monday.
France wants to limit any British right to interfere in eurozone decisions, and ensure Britain commits to allowing further integration of the euro area, a diplomat close to the negotiations said.
Last week, France secured some amendments it said were needed to ensure that efforts to balance the eurozone’s “ins” and “outs” did not give the City of London an unfair advantage.
Cameron hopes to agree a deal ahead of the talks to pave the way for an "in-out" referendum, likely to be held in Britain by June.
British officials struck a much more optimistic tone after Cameron’s meeting with Hollande.
A Downing Street spokesman said that the two leaders agreed that a draft proposal to reform Britain's membership of the EU would serve as a "firm basis" for a deal.
"They agreed that we are making good progress on the UK renegotiation and that the draft text from the European Council provides a firm basis to reach agreement at this week's summit," the spokesman for Cameron's office said.
Cameron, who is facing a barrage of criticism from eurosceptic and the right-wing press at home, will continue discussions with members of a wary European Parliament on Tuesday.
He has said that a current draft deal by EU President Donald Tusk will give Britain the ability to reduce immigration from the rest of the EU, protect its sterling-based finance industry, and ensure Britain’s right not to be drawn deeper into a political union in Europe.
Britain, which does not use the euro, wants to renegotiate the terms of its membership to ensure it never has to pay into a fund to protect the eurozone. It also wants any issues that affect the entire bloc to be discussed by all 28 member states, and not just the 19 nations that use the single currency.
Meanwhile, Tusk urged compromise to keep the bloc together during a visit to Bucharest. “This is a critical moment,” Tusk told reporters after a meeting with Romania’s President Klaus Iohannis.
After himself meeting Hollande in Paris on Monday, Tusk said he hoped for a deal in Brussels at the end of the week.