And according to The Sun, West Ham, Watford and Sheffield United are keeping tabs on the striker.
Celtic are also believed to be keen on Adebayor, who has not completed 90 minutes of football since January.
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West Ham are looking to replace departing striker Andy Carroll, while Sheffield United could opt for experience in their first season back in the Premier League.
Adebayor scored just three goals for Istanbul Basaksehir as the club finished two points behind Super Lig champions Galatasaray.
It is understood that the Togo international wants to play in London, which would put West Ham in the driving seat for his signature.
He made the unlikely move to Crystal Palace in 2016 and scored one goal in 12 appearances for the Eagles.
Now, he leads the line for Basaksehir and he has scored two goals in 13 appearances.
His overall record for the club is better, however, with 23 goals in 53 league appearances.
Gael Clichy
The right-back spent 14 years in the English top flight with Arsenal and then Manchester City.
Clichy now plays in Turkey
He made his name as one of the finest full-backs in the land with the Gunners and later made a £7million transfer to Manchester.
Now, he has been a solid outlet on the left-side of defence in his side’s impressive league campaign – netting once in 17 matches.
Emre Belozoglu
The Turk moved to Newcastle in 2005 and spent three seasons with the Magpies.
He appeared 58 times and once scored the winner in the Tyne-Wear derby.
Ultimately his career at Newcastle was hampered significantly by injury.
Emre has scored 14 times for the Super Lig side
Ten years on and two clubs later Emre moved to Istanbul Basaksehir – and the captain has scored 14 goals in 108 appearances in his four year stint with the club.
Eljero Elia
The Dutch winger moved to Southampton on loan in January for the rest of the 2014/15 season.
He made a good impression for the Saints and made clear his desire to sign permanently after scoring a brace against Newcastle.
Elia has made 13 appearances for Basaksehir this term
However, a deal never went through and he moved onto Feyenoord the following season.
This season he has one goal in 13 appearances for Basaksehir.
Demba Ba
The striker started his Premier League career with West Ham after moving from Hoffenheim in January and finished as their top scorer despite only playing for half the season.
He then became one of the best strikers in England at Newcastle United – scoring 29 goals in 54 appearances and earning a move to Chelsea.
Ba in action for Chelsea
For the Blues he scored some crucial goals but will be remembered mostly for scoring the goal after Steven Gerrard slipped to lose Liverpool the title.
Ba moved to Istanbul in January and has scored two goals in his opening four appearances for the league leaders.
Gokhan Inler
The Swiss midfielder was a rock in the Napoli midfield from 2011-2015 and moved to Leicester for their title-winning campaign.
He only made five appearances for the Foxes but that was enough to earn him a Premier League medal.
Inler has one goal in 27 appearances for the Turkish club
He struggled to break into the line-up due to the good form of N’Golo Kante and Danny Drinkwater and he moved onto Besiktas the following season.
After a year with Besiktas, Inler signed for the Turkish club and has one goal in 27 appearances.
Robinho
The Brazilian completed a stunning £32million deadline day transfer to Manchester City which would see them smash their club record fee for a player.
He enjoyed a good first season in the Premier League, notching up 14 goals and finishing as City’s top goalscorer.
Robinho playing for Manchester City
However, his second season was disastrous and he left the club to return to his former club Santos in Brazil.
Burnley marked their long-awaited trip to the continent after holding Istanbul Basaksehir to a goalless draw as Joe Hart managed to keep a clean sheet on his Clarets debut.
After getting past Aberdeen in the second qualifying round, Burnley were making their first trip overseas in 51 years and secured a stalemate at the Fatih Terim Stadium.
England international Hart made a couple of decent stops just 48 hours after signing from Manchester City but it was a low-quality contest that leaves the tie in the balance ahead of next week’s second leg at Turf Moor.
With an eye on Sunday’s Premier League opener at Southampton, Dyche made six changes from the side which edged Aberdeen, with the likes of Phil Bardsley, Charlie Taylor and Jon Walters all involved.
His side were slow to find their feet in the first half, at times literally as a series of players slipped on the pitch.
Basaksehir’s Eljero Elia came closer than anyone to making an early breakthrough but scuffed his shot at the crucial moment.
Emre, the 37-year-old former Newcastle playmaker, was directing much of the play but Burnley’s established defensive pairing of Ben Mee and James Tarkowski were letting little escape their attention as £15million signing Ben Gibson looked on from the bench.
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The pair also combined in the final third as the visitors made a rare foray upfield, Mee nodding down a hanging free-kick and Tarkowski hesitating over a shot he might have addressed more instinctively.
Hart’s first notable save for his new club came on the half-hour, responding well after Junior Caicara evaded Taylor and aimed low at the near post.
There was time for a second before the break, tipping over the crossbar after Manuel da Costa steered a corner goalwards.
The stalemate continued in a scrappy start to the second period, Jeff Hendrick closed down by goalkeeper Mert Gunok when one more cutback would have given Ashley Barnes a simple shot.
On the hour Basaksehir substitute Milos Jojic went close with his first touch, getting away from his man and seemingly beating Hart only to see his effort crash into the side-netting.
Wales striker Sam Vokes was sent on to offer another option up front but when Hart was booked for time wasting it seemed clear Dyche’s men were settling for what they had.
The hosts failed to create any more havoc in the closing stages, meaning everything remains up for grabs.
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