Nationalist Norman MacLeod wants the city council to fund the cost of feeding almost 16,000 youngsters.
The Pollokshields councillor said: "I have been told it would cost £2.9million and while that sounds like a lot of money it is peanuts compared to the council's budget of £2.1billion.
There is an enormous beneficial outcome to young people from a universal, non-means tested approach to the provision of healthy school lunches, which includes promoting sociability and increasing educational outcomes.
"In every instance where this idea has been tried, the number of young people remaining for their lunch increases materially."
Councillor MacLeod insisted free meals should be available to all children, regardless of how much their parents earn.
He said: "No matter how wealthy or poor they are, people go to the doctor or hospital where care is provided free. We don't charge wealthy parents who send their children to school."
Mr MacLeod first raised the issue of free lunches for primary pupils in April when the council set its budget for the year.
The move was rejected by the ruling Labour group, but the SNP councillor is planning to raise the issue again at a meeting of the council on Thursday.
He said: "I don't care who gets the credit for this - I just want it to happen.
"There was a trial of a similar scheme in Scotland in 2007/08 and the overall take-up of school meals increased from 53% to 75%. When you make free meals universal it removes the embarrassment for children who already receive free meals and maximises the number of people who get a healthy meal."
At the end of the last school year, Glasgow had about 15,800 pupils in primaries one to three. Of those, just less than one third were entitled to free school meals.
Stephen Curran, the council's spokesman for education, said: "The council already provides free breakfasts and free lunches to the children who need it.
"The millions of pounds this proposal would cost would mean a massive cut in all our schools.
"We would have to cut 95 teachers or cut 200 pupil support assistants to pay for it."