By Graeme Bailey Follow on Twitter
Ayes is out-of-contract at the Stade Velodrome this summer, and has spoken with a number of Premier League clubs.
Arsenal and Newcastle were among those strongly linked, but the 25-year-old has now flown into south Wales to undergo a medical.
Sources close to negotiations have confirmed to Goal that the Swans are confident a move will be finalised, although it is understood competition for his signature is strong.
Swansea lost star striker Wilfried Bony in January and Swansea boss Garry Monk will be hoping Ayew's arrival will just be the start of busy summer at the Liberty Stadium.
Ayew, who has been with Marseille for 10 years after joining their academy as a 15-year-old, is free to move on a Bosman free transfer.