The decision to issue a summons was made at Wednesday morning's meeting of the London Assembly's GLA oversight committee.
The Assembly wants to see any notes, text messages and/or social media messages sent or received in the time period 1 March 2018 to 25 January 2019 between the Garden Bridge Trust and/or the trust's professional advisors and the following officers of Transport for London:
• TfL Commissioner, Mike Brown
• TfL Head of Corporate Affairs, Andy Brown
• TfL General Counsel, Howard Carter
• TfL solicitor, Charles Ritchie
• TfL Director of City Planning, Alex Williams
The committee has also requested the disclosure of two letters sent by Lord Davies, chair of the Garden Bridge Trust.
The documents and files are to be provided no later than two weeks from the date of the summons notice.
Tom Copley AM, chair of the Garden Bridge Working Group at City Hall, said: "It is jaw-dropping that TfL have not learnt that when it comes to this project, transparency is paramount.
"We expect full cooperation with our continued investigations into the failed Garden Bridge project and will use all of the powers at our disposal to get it.
"Hiding behind the veil of 'commercially sensitive information' won't wash – when documents can be redacted, if necessary, before we place them in the public domain."
A TfL spokesperson said: "We have always supported the scrutiny of the Garden Bridge undertaken by the London Assembly and others such as the review conducted by Dame Margaret Hodge and will respond to this request as soon as possible."