The two options appear in the consultation on powers the government said were needed to update Sunday shopping laws written in 1994, long before shops faced competition from online retailers.
At present, stores with a floorplate in excess of 280 square metres may open for only six hours on a Sunday, though smaller stores can open all day.
The proposed powers would allow councils to opt to permit shops to remain open for longer, though they could disapply this for out-of-town stores so as to encourage commerce in town centres.
There would not be any power for councils to reduce existing Sunday trading hours.
Communities minister Brandon Lewis said: “We want to give local leaders the power to decide whether Sunday trading is right for their area, and to give their retailers the option to stay open for longer.”
Under the options, powers over Sunday trading could be available only as part of devolution deals, as an inducement to councils to agree these, though another option would give the power to any council.
The proposed extended hours were welcomed by business leaders including Dixons Carphone group chief executive Sebastian James, who said: “We live or die by whether we deliver for our customers, and if a community would like more time at the weekend to choose a new TV, phone, fridge or computer, then it will be good for our business to be allowed to offer that choice.”
But Association of Convenience Stores chief executive James Lowman said: “The government has turned its back on thousands of independent retailers, many of which will now be under threat of becoming unprofitable if changes to Sunday trading laws are made in their area.
“The consultation process for these reforms has been shambolic and opaque.”
The consultation runs until 16 September.