The city hopes to open all 26 of its libraries on Saturdays and as many as practical on Sundays.
The move will bring the city into line with many other major cities including Glasgow and Manchester.
The council said it was responding to growing demand from parents and other users.
But the move may be resisted by trade union Unison who say they have seen no evidence of public demand.
They also voiced fears that it would lead to other council services opening on weekends with an inevitable effect on their members working hours.
The proposed extra opening follows a surge in library membership in recent months, put down to people looking to borrow instead of buy books during the recession.
Councillor Deidre Brock, the city's culture and leisure leader, said: "Weekend opening is a great help, for example, to people who find it difficult to fit a library visit around their working hours during the week.
"It's quite unusual for a city of Edinburgh's size not to have weekend opening in its libraries and research in comparable cities shows that library users are delighted by this flexibility."
The council said that a 2008 review into the library service found significant demand for longer opening hours in a number of locations across the city.
Six city libraries currently open all day on a Saturday and on Sunday afternoons. The local authority is consulting with staff and unions on proposed shift changes to allow the extra weekend opening.
But John Ross, service conditions convener at Unison, said: "We can understand the concept of extending opening hours across the weekend but there is no evidence whatsoever that there is a public demand.
"There are currently 20,000 users a week and I am unaware of more than a handful that want this extension to Sunday opening.
"We are facing a 90 million deficit and if you extend library opening to a seven day week how do you get the money?
"We are in the middle of discussions with the council on pay models, which impacts weekend working payments and shift allowances and these discussions are ongoing. I do not see us agreeing library opening hours amendments ahead of that."
He added: "If the council manages to open libraries more at the weekend, they will start opening housing offices and such like as well."