With the significant progress made in the development of cathodes in lithium-sulfur (Li-S) batteries, the stability of Li metal anodes becomes a more urgent challenge in these batteries. Here the systematic investigation of the stability of the anode/electrolyte interface in Li-S batteries with concentrated electrolytes containing various lithium salts is reported. It is found that Li-S batteries using LiTFSI-based electrolytes are more stable than those using LiFSI-based electrolytes. The decreased stability is because the N-S bond in the FSI-anion is fairly weak and the scission of this bond leads to the formation of lithium sulfate (LiSOx) in the presence of polysulfide species. In contrast, in the LiTFSI-based electrolyte, the lithium metal anode tends to react with polysulfide to form lithium sulfide (LiSx), which is more reversible than LiSOx formed in the LiFSI-based electrolyte. This fundamental difference in the bond strength of the salt anions in the presence of polysulfide species leads to a large difference in the stability of the anode-electrolyte interface and performance of the Li-S batteries with electrolytes composed of these salts. Therefore, anion selection is one of the key parameters in the search for new electrolytes for stable operation of Li-S batteries.