A roll-out of 20mph zones to all residential and shopping areas in the capital – the most significant yet in Scotland – has also been proposed, together with a low-emission zone.
Under the radical move lorries and buses with inefficient engines would be charged for entering the city.
Environmental and sustainable transport groups hailed the plan, saying it would boost walking and cycling and safety in the capital, and cut pollution.
But the proposals, which will now go out to consultation, faced criticism from motoring groups who said the regime risked driving visitors away from the city and damaging its economy.
The Freight Transport Association said the low-emission zone could lead to price rises in shops as hauliers pass on costs.
The Automobile Association warned an attempt by Westminster City Council to introduce a Sunday parking charge was withdrawn in a hail of criticism. Churchgoers branded it a "tax on religion".
In the report, the council's communities services director, Mark Turley, said parking rules were drawn up before Sunday trading became widespread.
He wrote: "Free Sunday parking may not be maximising the economic benefit to retailers and other businesses – not all of the parking is by customers, and the lack of controls can reduce the 'turnover' of parking spaces.
"Also, the relaxation of parking restrictions leads to buses and general traffic experiencing delays on some routes on Sundays."
Profits from Sunday parking would be reinvested in bus services, while the impact on activities, "notably worship", and the impact on the city-centre economy would be considered before changes were introduced, according to the report.
As well as a blanket roll-out of 20mph zones, the report also suggests extending 30mph zones, although dual carriageways would remain 40mph.
Council transport convener Lesley Hinds said: "If we find that there is an appetite for looking at some kind of parking controls on Sundays, there's still a lot of work to be done to see how this would be implemented.
"We'd also need to evaluate the current parking patterns to see how parking bays are being used on Sundays. As with all parking restrictions, the aim is to keep traffic moving smoothly across the city and ensure that stationary vehicles aren't causing an obstruction or safety risk."
Ian Craig, managing director of publicly-owned Lothian Buses welcomed "this clear commitment to sustainable, green and prioritised public transport".
But Professor Stephen Glaister, director of the RAC Foundation, said: "It seems perverse to essentially tax the capital's motorists further still when the money will not even be spent on something like road maintenance."