It says that the three have all demonstrated substantial international experience on similar projects and the capability to complete complex works, required to build a bridge over a working river.
BAM Nuttall, a Bouygues TP/Cimolai joint venture and Dragados have all been shortlisted for the contract to complete the detailed design and construction of the Garden Bridge, which would have 270 trees including 45 different species planted and see approximately seven million visitors per year.
The selection included detailed submissions made by a variety of UK and international contractors.
The trust, a registered charity, cleared the third and final planning hurdle on Friday when the London mayor Boris Johnson’s deputy for planning Sir Edward Lister decided that he was happy for the decision to rest with the two planning authorities on either side of the bridge, Westminster and Lambeth, which both already given consent..
The mayor of London, Boris Johnson said: "The Garden Bridge will provide a fantastic new landmark for London whilst supporting regeneration and economic growth on both sides of the Thames.
"It will create a stunning oasis of tranquillity in the heart of our city and boost our plans to encourage walking in the city."
The bridge will include a major new public space and garden that will be free to access and will feature 270 trees as well as shrubs, climbing plants, hedges and flowers. It will be fully accessible, with two lifts at each end and will be open from 6am to midnight.
Transport for London and the Treasury have both committed £30m each with the remainder of the cost made up of private donations. The construction contract is estimated at around £90m with the total project cost estimated at up to £175m.
The bridge has drawn mixed reviews from organisations, with some including Royal Botanic Gardens Kew and English Heritage supporting the proposals and others, including St Paul’s Cathedral objecting on the basis that it will damage London views.
Vice chair of the Garden Bridge Trust Paul Morrell, who is also a fellow for the Royal Institution of Chartered Surveyors said: "The Garden Bridge will showcase the best of UK design, engineering and landscape talent.
"We have given careful consideration to the experience of all those who submitted proposals and believe this list gives a good selection of potential partners to realise this ambitious scheme. It will be a great bridge for a great city."
The final contractor will work with a range of specialists, Arup who are leading the design team with Heatherwick Studios as lead designer, Dan Pearson as planting designer Arup as engineers and technical specialists.
The trust expects to appoint a lead contractor in late spring 2015 with construction expected to start in the autumn. The trust believes the bridge can be open to the public in summer 2018.