The Premier League club made the application to Liverpool City Council on Wednesday that would, if agreed, increase capacity in both its main and Anfield Road stands.
The application is likely to be considered by the city's planning committee later this year.
A statement on the Liverpool FC website read: "If consent is granted, the proposals would see the capacity of a new Main Stand rise by 8,500, taking it to nearly 21,000.
"The outline proposals for the Anfield Road stand provide for an increase in its capacity of around 4,800 seats, taking the total stadium capacity to 58,800.
"The Main Stand would open onto a wide public space which has been designed to link the stadium and park while creating an area which is relevant and usable to the local community, particularly on non-matchdays.
"As part of its proposed stadium expansion plans, the club has also appointed Tom Doyle, an experienced project director who worked on the London 2012 Olympic programme, to manage the redevelopment should consent be granted."
Managing director Ian Ayre added: "We have the support from local residents and businesses for our proposed plans, that we can acquire the land required and that we can navigate the planning landscape.
"Over the past 18 months we have worked closely with the community, local residents, Liverpool City Council and Your Housing Group and we are delighted to be able to bring forward a planning application at this stage."
The club statement continued: "Construction work on the proposed Main Stand could begin early next year and it could be complete in time for the 2016-17 season."
Mayor of Liverpool Joe Anderson said: "The proposed redevelopment of the stadium is a hugely important part of our vision for Anfield.
"The overall regeneration of the area will see £260m invested and will deliver hundreds of jobs."