City council leader Andrew Burns said negotiations were now at “the end-game stage”.
And he revealed a signing ceremony could take place at the end of this month or the beginning of next.
The official UK government Budget document earlier this week spoke of “good progress” towards the deal, which will cover Edinburgh, the three Lothian councils, Fife and the Scottish Borders, along with university and business sectors.
Funding would come equally from the Scottish and UK governments and would be expected to lever in around £3bn of extra private investment.
Cllr Burns said: “There are very detailed negotiations going on between the two governments and the city region partners and we’re cautiously optimistic we will see very significant progress literally within a couple of weeks.”
A full list of projects under discussion has never been revealed, but the deal could include cash for new roads, the tram extension to Newhaven, and investment in research, education, tourism and culture, as well as addressing the wealth gap between communities.
New powers for the councils could include a tourist tax to be levied on hotel guests and a land commission to co-ordinate publicly-owned sites to help tackle the housing shortage.
The city council has previously said it is essential that public sector landowners such as the NHS, Police Scotland and the Ministry of Defence make land available for “accelerated development of low-cost and affordable homes”.
The city region deal will need to be approved by all six councils involved before it can be announced.
Cllr Burns said in Edinburgh a special full council meeting could be called in late March or early April and the other councils would meet around the same time.
But discussion of the deal will take place behind closed doors so the details can be kept under wraps until the official signing, which he said was likely to be held in Edinburgh two or three days after all the councils had approved the package.
Cllr Burns said: “There is a degree of opaqueness because there has to be that privacy about the final discussions in terms of the detail.”
Deputy council leader Frank Ross said all elected members of all parties in the council had been fully briefed on details of the deal, and he promised there would be public consultation on specific schemes at a later stage.
He said: “We’re signing effectively a heads of terms agreement on the big project. Within the big project headlines there will be detailed projects coming forward over the next ten years and those will be consulted at a much greater depth.”
Cllr Burns added that any large infrastructure projects would also have to get relevant planning permission.