The former assistant curator of the RIBA Drawings Collection will succeed Margaret Richardson to the position at the end of this month.
The appointment still has to go before the museum's trustees but it is understood that this will be a formality.
While at the National Trust, Knox was considered largely responsible for the acquisition of Tyntesfield Hall, the Gothic revivalist house near Bristol.
He was also involved with such projects as the restoration of Stowe Landscape Gardens, the acquisition of The Workhouse - a grim early 19th-century 'house of industry' at Southwell, Nottinghamshire - and the rescue of the Darnley Mausoleum in Cobham Park, Kent.
Knox was also chairman of the Mausolea and Monuments Trust between 2000 and 2004, and is a trustee of the Spitalfields Trust.