The move is designed to help Royal Mail better compete with rivals such as DPD and Hermes, which have announced plans to deliver on Sundays.
Royal Mail, which was privatised last year, said the initiatives would allow "shoppers not at home during the day to get their parcels".
Delivery offices with the highest parcel volumes across England, Wales, Scotland and Wales and Northern Ireland will open on Sunday afternoons, while Parcelforce Worldwide’s Sunday delivery service will launch in June for online shoppers buying off particular ecommerce sites.
The group is also to open its distribution network later on Saturdays and Sundays so that large e-commerce retail customers can post items ordered by consumers on a Saturday afternoon and Sunday for delivery on Monday.
Moya Greene, Royal Mail Group’s chief executive, said: "Through these new Sunday services we are exploring ways to improve our flexibility and provide more options for people to receive items they have ordered online.
"The support of the Communication Workers Union has enabled us to respond quickly to a changing market, underlining the importance of the ground-breaking Agenda for Growth agreement."