The SNP had promised to abolish the charge when it was first elected to power in 2007, with ministers forced to go back on this two years later as there was “no consensus” in Holyrood for the change.
Since then the Scottish Government has made some reforms to the charge.
With the SNP now forming a minority administration, Greens believe they can build on the work of the Commission for Local Tax Reform, which recommended axing the tax in 2015.
Finance secretary Derek Mackay insisted the Scottish Government was “committed to making local taxation more progressive”, adding he was “open to further dialogue on options for local tax reform”.
The Greens have already told SNP ministers they want changes to be made if they are to support the government’s budget next year. Andy Wightman, the party’s local government spokesman, said that at the moment the public was “being penalised with an outdated, unfair system under which most households are paying the wrong amount”. He added: “We have a chance to build on the agreement struck during the 2015 cross-party Commission on Local Tax Reform that the present council tax system must end.”