The 24-year-old’s proposed move to appeared to have stalled last week, but the Super Lig side rekindled their interest over the weekend with a loan deal until the end of the season now set to be finalised, with the possibility of a permanent deal in the summer.
Mallan has scored four goals in 21 appearances this season, but the former Scotland Under-21 star has seen his starting opportunities limited further following the arrival of Jackson Irvine and Chris Cadden.
Ross said: "Stevie travelled to Turkey yesterday, arrived very late last night. I think he has now completed all his necessary medicals and as I understand it, I think it's just subject to international clearance, so all being well, that should be announced in the near future.
"It's an interesting loan move for him, a very different opportunity for him as well. It's something that's obviously come about quite quickly after it looked like the opportunity had passed away last week."
Both Kevin Nisbet and Ryan Porteous trained with Hibernian on Monday after the club rejected bids from Birmingham and Millwall respectively last week.
Ross was unaware if Birmingham had launched a fresh bid on Sunday, adding: "I've not been informed otherwise over the last 24 or 36 hours, other people at the club would probably answer that better than me.
"But from the football side of it, he trained this morning and, as things stand, will be with me tomorrow evening for the game at St Mirren."