SSE made the offer to city council leader Gordon Matheson during an energy summit aimed at helping people struggling to pay their fuel bills.
Prepayment meters are used by many suppliers to allow customers to manage their use of electricity or gas, or when their accounts are in arrears.
However, they have caused controversy, with householders saying they are unable to access discounted tariffs available to those who pay by direct debit.
The only way they can benefit from the lower prices is by having the meter replaced, which can cost around £60.
It is hoped SSE's pledge to drop the cost of removing prepayment meters will be the first in a series of measures designed to tackle fuel poverty.
Suppliers have agreed to work with the council to share information on energy use which will help identify those in need and to investigate how the best tariffs can be offered to city customers.
Mr Matheson said: "The rising cost of energy is a very serious concern for many people, particularly those on low incomes. The prospect of having to find extra money just to replace a prepaid meter is a very significant barrier to some of our most vulnerable citizens getting the best deals.
"I am encouraged SSE recognises that problem and is serious about working with us to give customers in Glasgow a fairer deal."
SSE's managing director for enterprise, Jim McPhillimy, said: "We recognise our responsibilities to our most vulnerable customers. While we believe prepayment meters still have an important role to play, with many customers preferring this option as a way of budgeting energy costs, we recognise that in some cases it is not suitable for a customer's circumstances
"We will work to identify customers for whom a prepayment meter is no longer suitable and waive the cost of removing the meter, when it is appropriate and the customer can meet their ongoing costs."
Under the new deal, the energy firm will waive charges when its customers are referred by advisors at Wheatley Group, which includes Glasgow and Cube housing associations, or the city's home energy advice team G-HEAT.
Mr Matheson is now trying to persuade other energy firms to follow the lead of SSE. He said: "We won't always agree, but the energy firms have come to the table and I want to keep them there and press them to do everything they can to help their customers in Glasgow."