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It is understood the unnamed company approached the council around 18 months ago, but has not had a positive response.
The council's housing committee has since agreed to invite investors, developers and landowners to bring forward opportunities for the council to accelerate provision of affordable housing, but critics say conditions laid down make a lease deal virtually impossible.
Former SNP housing convener Gavin Barrie said he feared politics were being put before finding homes.
He said he was in touch with the company involved and it was still interested in doing a deal.
Cllr Barrie, who quit the SNP and sits as an independent, said: “It's an opportunity other councils are taking up whereby private finance purchases homes and lease them to the council over a long period of time. The lease would be covered by the rents and if people were on benefits the Housing Benefit would cover the lease fees. And at the end of the lease period the houses would revert to the council's ownership.
“The council has now had this deal offered to them for about 18 months and at the last housing committee they agreed to go out to the market to see what was there and act on it in about six months’ time.
"I understand officers have to be seen to be even-handed across the market, but Edinburgh has a housing crisis, has had for years, and there seems to be a reluctance to explore new ways of providing housing and doing that quickly with the urgency our housing crisis deserves.”
He said he understood people might be suspicious of private finance companies being interested in affordable housing, but he said they were the people with money to invest.
“We deal with private companies all the time. The council doesn't have a house-building department – private companies build houses for the council, so it’s not as if we're not involved with business or investors in building affordable housing.
“If people have political point of view, they are entitled to that but let's get people housed first and then we can argue about the politics once everybody’s got a home.
“This lease-type deal has been taken up by other councils and they have obviously carried out due diligence on it and believe it to work. I've had no explanation as why it would not work for Edinburgh.
“Had they acted on this 18 months ago, people could have been in houses by now."
But housing convener Kate Campbell rejected the idea there was any political resistance to the idea.
She said the council could not just hand one company a contract of such a scale without going through a proper fair and transparent process, allowing other companies to bid.
"We have agreed to go ahead with a procurement process, which is about leasing but also if people want to sell us land that's something we're interested in as well. It's about increasing the number of different types of affordable housing.
"We're embracing the opportunity, but in a way that's fair and transparent."
And she denied the council had set impossible conditions.
"We need to test the market and do it in a way that gets best value for the citizens of Edinburgh and make sure we're delivering as much affordable housing as we can."