Councillor Donald Anderson, the city's sports and leisure leader, told around 600 protestors at a public meeting on Saturday that the council would like to rebuild the ageing stadium - if it had the money.
He also said a steering group, made up of local residents, sporting organisations and the council, should be established to help decide Meadowbank's future.
At the moment, the council plans to demolish Meadowbank and sell the site for housing in order to fund a new sports complex in the west of the city, which would be built by 2011.
City leaders say it would be too expensive to refurbish Meadowbank, although a new, smaller, sports centre would be built on the London Road site.
Cllr Anderson came under fire at the meeting, with protestors accusing the council of making the decision to demolish Meadowbank behind closed doors.
They claimed that knocking down the stadium would "rip the heart out of the community".
Cllr Anderson told the meeting: "If I had a magic wand and a blank cheque, we would demolish Meadowbank and build a new stadium on the site. We can, and I'm sure we will, go back to sportscotland to see of there is any more money that can be squeezed out.
"The best option for keeping Meadowbank is to demolish it and rebuild it from scratch. The question is still open. It's still open for the council to decide how to handle this."
Edinburgh East MP Gavin Strang today demanded a "radical rethink" of the council plan. He said the huge turnout at the meeting demonstrated the strength of feeling on the issue.
"The proposal will have to be fundamentally reconsidered," he said. "They have got to realise we cannot use Meadowbank as a cash receipt to finance the grandiose plans for Sighthill."
Protesters and members of the Save Meadowbank campaign believe Cllr Anderson's comments are a step in the right direction.
The campaign has attracted more than 5000 signatures on petitions and has received the backing of a number of high-profile celebrities, including Olympic champion Allan Wells, The Proclaimers and Radio One DJ Edith Bowman.
A consultation is currently under way asking for comments and ends at the end of the month. Save Meadowbank campaigners are planning a march to the City Chambers on March 30 to deliver their petitions.
Campaign spokesman Kevin Connor - who revealed he might stand in the forthcoming elections - said: "I think it was a fantastic meeting and we really achieved the objectives we set ourselves.
"We went in there with the intention of putting pressure on the politicians and make them aware of the immense strength of feeling and that they may have to rethink their policy with this issue.
"Now people are starting to understand that the consultation is phoney and pre-supposes the demolition of Meadowbank."
FOO FIGHTERS AND KAISER CHIEFS SUIT TO A T
THE largest shows in Fringe history will form the centre of the T On The Fringe music line-up.
Capacities of 25,000 have been announced for performances by rock legends Foo Fighters and chart-toppers Kaiser Chiefs at Meadowbank.
The number of tickets available for the gigs is 3000 higher than the previous record - when 22,000 were sold for Snow Patrol last year. T On The Fringe organisers are keeping the rest of the line-up a secret. Venues are expected to again include the Liquid Room, Corn Exchange and Cabaret Voltaire.
Foo Fighters will play on August 21, with Kaiser Chiefs three days later. Tickets, priced 37.50 and 28.50 respectively, go on sale on Wednesday.