The North Macedonian, at present, is set to leave Elland Road at the end of the season, four years after joining from Swiss side Lugano.
During that time, Alioski has become a terrace favourite, playing a big part on and off the pitch in the Whites’ successful Premier League promotion charge.
The 29-year-old was first linked with Galatasaray in January – and reports have resurfaced over the last 24 hours.
As per Turkish outlet Milliyet, Alioski is said to have rejected Leeds’ latest contract offer and is ‘very close’ to joining Gala.
Alioski has started the Whites’ previous four matches on the bench, fueling speculation over his future.
YEP chief football writer Graham Smyth tweeted in response to the rumours: “Hasn't rejected an offer as far as I'm aware. Nor has he signed it, however.”
Meanwhile, former England and Leeds goalkeeper Paul Robinson has urged his former club to hijack Leicester City’s move for Celtic striker Odsonne Edouard.
It was reported last week that the Foxes were closing in on a £15million deal, who would reunite with Brendan Rodgers.
However, Robinson wants to see Leeds, who were linked with the 23-year-old last summer, join the race and supply competition for Patrick Bamford.
Robinson told MOT Leeds News: “I think there’s a need in the summer to bring a striker in.
“£15m for Odsonne Edouard is a really great deal. If anyone can get him for £15m it would be amazing.
“I’d love to see Leeds hijack that deal. Edouard can play as a nine and he’s strong. He’s a player who can also play behind the striker.
“If you can get him for £15m… I’d like to see Leeds hijack that deal but I think a lot of other clubs will be in for him too.
“He won’t be on Premier League wages either at Celtic. It won’t take a lot to buy out his contract.”
Here are the latest transfer headlines from Leeds and their Premier League rivals: