Senior councillor Trevor Davies is leading the bid to outlaw the weapons by introducing a special by-law. And supporters say they believe the move will have cross-party backing on the council.
The plan was revealed as a woman from West Lothian whose brother was killed by an airgun gave her backing to calls for a nationwide ban.
Jacqueline Jack, who has collected thousands of signatures on a petition supporting MSP Tommy Sheridan's Holyrood Bill to ban airguns, said: "I don't want anyone else to go through what we've gone through."
Police are being called to nearly five airgun incidents every week in the Lothian and Borders area.
And there is also mounting concern about the number of attacks on emergency workers, some of which have involved air weapons. Details of the proposed new by-law are yet to be worked out, but council officials are being asked to look at the maximum powers available.
Any by-law passed in Edinburgh would have to be approved by the Scottish Executive. The ban would then be enforced by the police.
Councillors in Aberdeen have voted in favour of a similar move. Councillor Davies said he was not out to ban airguns held for serious sporting reasons.
"We don't want to impact on that. It's when guns get into the hands of communities it's a problem," he said.
"We have had families devastated by damage to children from airguns and some people are converting them into genuine guns, mainly done abroad before they import them here. "Clearly there is the technology for this possibility and it's a worry. It's interesting to see a Labour councillor in Aberdeen who has taken the initiative through a by-law.
"If there is a way to do it before parliament then we should see if we can as a matter of public safety."
The latest statistics show a 46 per cent rise in airgun-related crime in Lothian & Borders over the last six years, with 244 incidents, up from 167 in 1999/2000.
Council leader Ewan Aitken said he thought most councillors would back a new by-law.
He said that in his own Restalrig ward a constituent had contacted him about her pet being shot at constantly and banning airguns would lead to greater safety throughout the Capital. He said: "Given reports about recent attacks by airguns it is right we should do something. People talk about the need for safer streets and this is one way we can go about achieving that.
"We need to work out the details but it is with reference to a by-law. We want to use our powers to maximum effect."