According to this R-sig-Geo forum question, it seems the Close gaps
tool uses the Nearest Neighbour interpolation method. The following is quoted from the webpage (emphasis mine):
Q. Would anyone know a good way to fill in holes within a DTM? There is no data inside these holes and it is affecting my calculations, so I prefer for an interpolation of the nearest neighbors to fill in a value or something similar.
A. There is a very efficient function in SAGA called "Close Gaps" that does exactly that. What makes it especially efficient is that it allows you to set a mask map.