Media planning and buying is managed by Carat London and M4C respectively.
Around ten agencies were sent RFI documents on Monday, with a shortlist expected to be announced at the end of September.
Agencies have been asked to submit information for both media planning and buying, but the review could evolve to cover media planning only.
18 Feet & Rising, which runs the ad account, is not affected. It recently created the "I oakleaf" print campaign, which played on the well-known "I heart NY" design.
The National Trust did not respond to requests from Campaign for comment.