Sir Christopher Wren used the chapterhouse, which was originally built in 1332, as a site office during the cathedral's reconstruction in the early 18th century.
The decision on whether to rebuild the historic structures - partly as a walled-off seating area - will be confirmed at a Corporation of London Planning and Transportation committee meeting next Tuesday (13 September).
The remains of the original octagonal chapterhouse and cloister court, which are Grade II*-listed, will stay partially covered by paving in the cathedral's southwest courtyard.
A 1973 statue of Sir Thomas Beckett which currently rests on an exposed chapter house buttress will be moved temporarily to the southeast churchyard.
Once the work is finished, the statue go to a new permanent home in the north churchyard, on the site of the 12th century Deanery chapel.