The Merseyside club revealed that development plans had been submitted to the city council on Wednesday.
If permission is granted, the Premier League side will increase the amount of seats in both the main and Anfield Road stands.
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A decision is likely to be made by the city's planning committee later this year, while the club hopes construction work will be completed in time for the 2016-17 season.
"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," read a statement on the club's official website.
"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 program, to manage the redevelopment should consent be granted.
"Construction work on the proposed Main Stand could begin early next year and it could be complete in time for the 2016-17 season."
Managing director Ian Ayre commented: "When we set out on the journey to explore the feasibility of expanding the stadium, we said the process would be determined by certainty. Certainty that 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 Mayor of Liverpool, Joe Anderson, spoke of his excitement, adding: "The proposed redevelopment of the stadium is a hugely important part of our vision for Anfield.
"The overall regeneration of the area will see 260 million pounds invested and will deliver hundreds of jobs."