The 28-year-old French frontman was tipped to join Newcastle, but we understand he is close to completing a move to South Wales following the successful conclusion of talks over personal terms.
Gomis, who is out of contract, was a target for the Swans last summer, but they could not get the deal across the line and turned instead to Wilfried Bony.
There has been speculation that Bony could leave the Liberty Stadium this summer with several clubs said to be monitoring is situation after a hugely successful first campaign in English football.
However, reports from South Wales have suggested that Swans boss Garry Monk wants Gomis to play alongside the Ivorian, rather than in place of him.
Gomis has been a long-term target for Newcastle, but talks between the Magpies and the player's representatives failed to broker a deal and the Welshmen appear to have made a decisive breakthrough.