Staff and operations at the centre in South St Andrew Street will be transferred to High Riggs or Wester Hailes.
The Broxburn jobcentre is also earmarked to close, with its work transferred to Livingston.
Citizens Advice Scotland said the closures would hurt low-income families because they could not afford to travel long distances.
Edinburgh North & Leith SNP MP Deidre Brock said: “Sending people from such an easy-accessed, central location to the other side of the city just doesn’t make sense. We need to break down barriers to finding work, not build new ones. I’ll be seeking a rethink.”