I finally found it!  I've tried to get this for years.  A link to an
Eliot portrait at the National Portrait Gallery (U.S. Smithsonian
Institute).  The link still isn't where it is supposed to be though.

A few years ago I can across this portrait in a room at the NPG that
also had a unique bust of Ezra Pound. The painting is about 4 feet
high and the titles of the books on the shelves are quite legible.
As I remember, at least one birding handbook was there.

This page has a small image and a description and a history of the

This page has a somewhat bigger image:

The following information about the painting came from using the
National Portrait Gallery's search engine at URL and then entering "Eliot" in the
"sitters" field and then, after submitting the search, selecting the
NPG link brought up.  The image on the resulting page is not found and
shows as a broken image icon.  Also the link from the broken image
does not work.

    Accession Number: NPG.86.88
    Sitter: Eliot, Thomas Stearns (b.1888,d.1965)
    Artist: Kelly, Gerald Festus (b.1879,d.1972)
    Classification: Painting
    Date: 1962
    Dimensions: 114.6cm x 94cm (45 1/8" x 37")
    Current Owner: National Portrait Gallery, Smithsonian Institution
    Acquisition: Purchase from
        Gift of Nahum, Peter
        Commissioners, National Portrait Gallery
        Senior Staff, National Portrait Gallery, May 1986

    Rick Parker