The Les Bleus international is set to out of contract at Lyon in a few days’ time, with the Ligue 1 club having failed to agree a new deal with him.
Newcastle were linked with a move for his services both last summer and over January, and were widely expected to snap him up this time around.
But now the player himself has opted for the Swans, and will make a trip to the club to undergo any formalities before penning a deal.
Reports suggest that the 28-year-old was offered £75,000-per-week by the Magpies, hinting that Swansea’s proposal is of a similar level.
Sunderland and West Ham have also been linked with Gomis, but neither club look likely to mount a serious late counter-offer.
Garry Monk’s side may face a battle to keep Wilfried Bony and Michu at the Liberty Stadium for next term, making the likely capture of the powerful Frenchman a potentially shrewd one.