Fletcher lived close to the Globe Theatre in Park Street, Southwark, sharing his life with fellow playwright Francis Beaumont.
The unknown artist shows Fletcher as a prosperous and well-dressed man with paper and pens. If the portrait is acquired it will be hung as part of a special display with Shakespeare and Ben Jonson. The picture is on offer for £218,000 and, although some funding has already been found, the Gallery must raise £50,000 by Sunday 20 January.
The portrait is on show in the Tudor Gallery at the National Portrait Gallery.
Donations can be made online here where further information are also available.
John Fletcher on Wikipedia.