The plan was for the council, which has been aiming to become net zero by 2030, to install air and water source heating systems at both sites and to insulate the roof at the leisure centre.
The council has spent more than £500,000 on looking into the viability of the scheme but has concluded it would not be possible to deliver the project on time and within budget.
A plan to insulate the roof at the Riverside Leisure Centre has been scrapped
The Department for Business, Energy and Industrial Strategy awarded Exeter City Council £6.392m, with a contribution of £871,000 required from the council.
A condition of the grant was that the money had to be spent by 31 March 2025.
Exploratory work carried out found there "significant challenges" to delivering the project at both sites, according to a report to be considered by the council's executive on Tuesday.
The report said there were "logistical constraints which render the scheme undeliverable" at the museum.
It said the additional electricity required for air source heat pumps would take between three and six years to provide.
The report said there were "a number of issues and concerns" relating to work at Riverside, including which areas would be affected and for how long and the extent of damage to the roof and the length of time needed for repairs.
Riverside had a new roof installed ahead of its reopening in 2021 following a fire in 2017, with an estimated budget for all of the works of £4.4m.
Dianna Moore said the was a "golden opportunity" to make Riverside Leisure Centre more energy efficient
Diana Moore, leader of the Green Party at Exeter City Council, said it was a "disaster" to learn the authority had planned to scrap the scheme.
She said: "I am really shocked that the council is throwing in the towel on £7m having spent £500,000 of its own money already on this project.
"It is a real loss in terms of reducing the carbon emissions from this building, which is a state of priority, and, by doing that, it would have reduced the running costs.
"Many pools are closing around the region and we have a golden opportunity to make this building more efficient.
"They can't give back the money in the hope that more money will come down the track. I think it's a complete disaster."
Exeter City Council said it hoped to recover the £553,000 spent so far from the government.
According to BBC.com