Willie Hart, the union's local secretary, described the proposal as "a false economy" and "an insult to the children of Glasgow".
Councillors will decide in January whether to go ahead with the plan to replace the nursery teachers with child development officers (previously described as nursery nurses in the authority).
The 37 establishments which would be affected all have headteachers and were considered large enough to support nursery teachers as well.
The EIS is warning, however, that the proposal could prove to be the "thin end of the wedge" and that the authority's next move might be to remove nursery heads and those nursery teachers who currently work in primary schools.
Mr Hart said: "The Scottish Executive talks today of a curriculum for excellence and of education 3-18. It stresses the importance of continuity through nursery, primary and secondary schooling and the need for smooth transition from one stage to another.
"A key part of that continuity and smooth transition is the planning and communication between teachers in the different sectors. Continuity of learning and smooth transition from nursery to primary will be much less likely if there are no teachers in nurseries.
"How well your child learns right up to secondary school and beyond will be greatly influenced by how well she or he was taught how to learn in nursery. The only staff fully equipped to do that by five years' minimum university and professional training are nursery teachers."
The union also contrasts Glasgow's proposals with provision in other authorities.
"All nursery school children in East Renfrewshire have two teachers in their nursery schools in addition to the headteacher," the institute points out. "In England, there is at least one nursery teacher per school."
However, Christine Stephen of Stirling University's Institute of Education - an expert on pre-five education who recently gave evidence to the Holyrood early years inquiry - suggested that the situation was not quite as clear-cut.
"Just being a teacher does not make you a great practitioner," Dr Stephen said.
Other pre-five staff were now entering the sector with early years qualifications at undergraduate degree level. More research needed to be done into which qualifications make a difference to children's experience, she said.
Another key factor was the quality of leadership given by the headteacher.
A spokeswoman for Glasgow City Council said: "A range of options is being considered. No decisions will be taken until early next year."