City of Glasgow, Dundee City, Inverclyde and West Dunbartonshire Councils are also hopeful they will hear within days that their separate pleas for extra government money this year have been successful.
All four argue that they were particularly disadvantaged by the review of boundaries in the 1996 reorganisation. Dundee and Glasgow, for example, are acknowledged to have a good case in the loss of a significant proportion of their outlying higher council tax band properties to neighbouring councils.
Craig Roberton, chair of finance at Glasgow and finance spokesman for the Convention of Scottish Local Authorities, said the city had been 'walking a tightrope where we have been making a case for ourselves, but not at the expense of others'.
Glasgow was expecting an announcement shortly, Mr Roberton said. 'Whether ministers will agree to all four cases is another matter.'
COSLA was due to meet Scottish finance minister Jack McConnell today to put flesh on its agreement to review the grant distribution mechanism with particular emphasis on those factors which take account of deprivation.
This review is expected to take at least 12 months to complete.
Scottish councils have also agreed to a short-term arrangement to target any new money, should it become available, at councils with the greatest financial difficulties.
Dundee leader Julia Sturrock stressed that all councils were facing continuing financial difficulties as a result of government policies such as the self-financing of pay awards.
'If their situation is difficult, ours is dreadful just because we have all their problems and then some,' she said. 'It has taken a long time for local government finance to become this unfair and unjust. We can't wait any longer for justice.'