The 25-year-old is a free agent, with his contract at Tannadice due to run out over the summer.
Swanson's agent John Viola told the Peterborough Telegraph: "Danny will travel to Peterborough this week to have his medical. If there are no problems he will sign a two-year contract with Peterborough."
Viola added: "Peterborough will have themselves a fine player. He could have signed for several other clubs, but he liked what Peterborough had to say and they have said they will play him in an attacking midfield role which is what he wants."