The centres run at a deficit of Pounds 500,000 and West Lothian has refused to contribute to their costs without a similar arrangement for Lowport outdoor centre in Linlithgow, which it took over in April.
John Dobie, depute director of education in Edinburgh, said he was "disappointed at the latest turn of events". The city council was accepting a "quite unfair burden", he suggested.
But Gerry Fitzpatrick, head of community and leisure in West Lothian, maintained: "We cannot continue to provide 100 per cent of the costs at Lowport and provide free places for children from other councils."
Last term, 233 pupils and teachers from West Lothian used Benmore and Lagganlia while 371 from Edinburgh, Midlothian and East Lothian used the residential facilities at Lowport. A further 540 used Lowport on a daily basis.
The four directors of education will meet on August 15 to find a way out of the impasse.