Edinburgh City Council said some families who had applied to popular schools had no chance of getting their child in.
Some schools are still taking placing requests, but numbers of successful applicants are likely to be small, it said. A report to the council's education committee next month will highlight growing pressure on places, with P1 registrations increasing by 3.5% on last year to 4632.
There has been a 15% decrease in the number of placing requests on last year, thanks to a campaign to encourage people to look at their catchment area.
However, convener of education, Paul Godzik, said the council had still received more than 700 placing requests so far.
He said: "Parents who have submitted a placing request should be prepared for the fact that they may not get the place they were hoping for, and recent experience has been that appeals are unlikely to be successful."