The 28 year old Turkish international striker is currently out of favour with Stoke City after their £8 million capture of Kenwyne Jones.
Stoke manager Tony Pulis is hoping to recoup some of the money splashed on Jones but is annoyed at having to part with Tuncay for less than the £5 million they paid Middlesbrough little more than a year ago.
Tuncay is said to be keen on a move to the Latics where he is likely to get first team football and play in a team that would suit his style.
Meanwhile, the departure of Jason Scotland is on hold as Martinez scrambles to add further attacking strength to the squad.
Scotland looks set to join Roy Keane at Ipswich as soon as the Latics add Tuncay, and Franco Di Santo to the squad.
Di Santo spent the duration of the previous season on loan with Blackburn, and the former Chelsea striker is keen on another move to the North West as he aims to add depth to his top flight experience.
At just 19 years of age, the transfer would represent another youthful target as Roberto Martinez continues to build a Wigan Athletic team for the future.