You can do this using R and biomaRt. Here's an example.

First lets create two example sets of SNPS. Between these the first pair are identical, the second pair are synonyms to each other, and the third pair are distinct.

snps1 <- c('rs4844600', 'rs4266886', 'rs6656401')
snps2 <- c('rs4844600', 'rs61737012', 'rs386638846')

Next we load the biomaRt package, and query the variation mart to return all the synonyms and their sources for our first set of rsIDs.

library(biomaRt)
## use the Ensembl variation mart
snp_mart <- useMart(biomart="ENSEMBL_MART_SNP", 
                    dataset="hsapiens_snp")

## get the synonyms and their source for our SNPs
results <- getBM(filters = c('snp_filter'), 
           attributes = c('refsnp_id','synonym_name','synonym_source'), 
           values = snps1, 
           mart = snp_mart)

For reference, the first few rows of results looks like the below. You can filter at this stage if you know you only have synonyms from a certain source.

> head(results)
  refsnp_id             synonym_name synonym_source
1 rs4266886               rs61198255  Archive dbSNP
2 rs4266886 NM_000651.4:c.487+787T>C     dbSNP HGVS
3 rs4266886 NM_000573.3:c.487+787T>C     dbSNP HGVS
4 rs4844600               rs58362463  Archive dbSNP
5 rs4844600               rs61737012  Archive dbSNP
6 rs4844600     NP_000564.2:p.Glu60=     dbSNP HGVS

We can now combine our original set of rsIDs with their synonyms.

snps1_complete <- c(snps1, unique(results$synonym_name))

and then ask which of our second list of IDs is in this expanded list. We see it finds two entries as expected.

> snps2[snps2 %in% snp1_complete]
[1] "rs4844600"  "rs61737012"