The new recruit will curate three shows a year, collaborate with artists and take charge of 5,500 rare horticultural objects.
The south London museum was the first in Britain dedicated to shows about gardens and their design. The new curator will replace Mary Guyatt, who left to study.
Museum manager Jess Turtle said: "The closing date is 20 December and we hope to appoint a curator in January. The response has been good and we should get someone excellent."
She said one of the principal strengths for the new curator would be a professional background in museums and exhibitions, but an interest in gardens was important.
The salary is £30,000pa and shows can cost up to £30,000 to host, said the museum, which drew 28,000 paying visitors last year. Next year's shows include one on Dutch masters such as Piet Oudolf.
Another, entitled From the Garden City to the Green City, will run throughout next autumn into 2012 and is currently at a conceptual stage, said Turtle. Other shows next year will focus on artist in residence Charlotte Verity, Tom Stuart-Smith and Dan Pearson.