They want the city chosen by the Scottish Government to pilot a ban on the most polluting vehicles to cut toxic fumes.
However, the move could put Edinburgh on a collision course with Glasgow, which is expecting to be the trailblazer next year.
Anti-pollution campaigners Friends of the Earth Scotland (FoES) said Glasgow was the front runner because the SNP and Greens, who are likely to control the city council, committed to a LEZ in their manifestos for last week's elections.
It said Edinburgh would have to "nail its colours to the mast quickly" if it wanted to in with a chance.
FoES said the SNP, which is also expected to lead Edinburgh City Council, had not included backing for a LEZ in its manifesto, although new leader Adam McVey has supported it.
He told The Scotsman today the SNP would be "strongly considering" the Greens' proposal.
Susan Aitken, who is expected to become Glasgow's new SNP council leader, said last month: "Glasgow is the obvious choice and I think the Scottish Government would welcome that."
However, Edinburgh Green councillor Chas Booth will call on the city council’s new administration to swiftly contact the Scottish Government to put the capital’s case.
He said it was a “matter of urgency” because government funding may be available only for the first LEZ.
In a motion to the first meeting of the newly-elected council next Thursday, he will urge the local authority to act fast.
The motion calls for the council chief executive to “urgently write to the Scottish Government offering to work towards establishing Scotland’s first Low Emission Zone in Edinburgh, with the expectation that the Scottish Government can make a significant contribution towards the cost of establishing the zone, and offering early talks to discuss the practical details of bringing this about.”
Mr Booth said nitrogen dioxide levels were exceeded at 26 monitoring stations across Edinburgh in 2015.
Levels of another deadly pollutant from vehicle exhausts, particulates, were exceeded at Salamander Street in Leith and three other sites.
Mr McVey said: “We [the SNP] pledged at the council election to take decisive action on air quality and tackle the number of the heaviest pollutant vehicles entering the city.
"Establishing an LEZ is a serious option to help Edinburgh make significant progress on air quality and we have no principled objection to this way forward.
"The SNP will be strongly considering the Green party’s motion in the lead up to first full council meeting.”
Ministers have pledged to establish the first LEZ by next year.
However, so far they have only said it would in one of Scotland's four largest cities - Glasgow, Edinburgh, Aberdeen and Dundee.
Transport minister Humza Yousaf told The Scotsman in February he had noted that Glasgow City Council's then opposition SNP group was committed to a LEZ for the city.
Mr Yousaf said: "If there is a LEZ, you cannot be half hearted about it.
"It could drastically reduce traffic."
The minister said the zone would be enforced using number plate recognition cameras to deter banned vehicles.
FoES said all four cities urgently needed LEZs.
Air pollution campaigner Emila Hanna said: “Edinburgh, Glasgow, Dundee, Perth and Aberdeen all deserve Low Emission Zones because they all have dangerously high levels of air pollution years after a legal deadline.
"The Scottish Government’s response to the air quality public health crisis, which is claiming over 2,500 lives in Scotland each year, has been woefully inadequate.
"It has promised just one LEZ when we know that all the big cities in Scotland have dangerous levels of air pollution.
“It must fund these low emission zones across the country to ensure that cash-strapped councils can deliver these vital improvements."
"Air pollution from traffic is endangering the health of some our most vulnerable citizens - the young, the elderly and people with existing illnesses."
A spokeswoman for the Scottish Government's Transport Scotland agency said: “We are determined to improve air quality and are working to ensure Scotland’s first low emission zone is in place next year in one of Scotland’s four major cities.
“The Scottish Government is liaising closely with local authorities and other partners to identify the first LEZ early adopter, following the local elections where LEZs were noted as a key environmental commitment in a number of manifestos.
“We are undertaking a review of both the regulations for enforcing LEZs, and the enforcement technology needed to operate a LEZ, and are also hosting nationwide engagement sessions with the bus and freight sectors around the principles of LEZs.
“We plan to launch a consultation on the national low emission framework in the summer, which will provide drivers with the opportunity to comment on the criteria for our first low emission zone.”