Via BEDOPS bedmap:

$ bedmap --echo --echo-map B.bed A.bed > answer.bed

The file answer.bed will contain each element from B.bed (each gene) and — adjacent to the gene — a list of variants from A.bed that overlap that gene. Add the --skip-unmapped option to remove genes from B.bed that do not have overlaps with variants from A.bed.

To run this with R, use bedr or system:

1. https://cran.r-project.org/web/packages/bedr/index.html

2. http://stat.ethz.ch/R-manual/R-devel/library/base/html/system.html

In addition to the suggestion from cpad0112 (note that bedtools/bedops in particular are probably faster than this) you can use the GenomicRanges package:

library(GenomicRanges)
a = c(a, subsetByOverlaps(b, a))

Again, I expect bedtools or bedops are a bit quicker, but you don't have that many regions, so it shouldn't much matter either way.