The 26-year-old was reportedly the subject of a £12.5 million bid from CSKA Moscow, with Rubin Kazan also said to be interested.
However, his agent claims Ajax is just one possible destination, while the player himself is keen for a summer switch to the Premier League with the Gunners and told Turkish media:
I am a Trabzonspor player till the end of the season, if I was to move abroad I would want to move to a big club, I would accept a transfer to Arsenal.
Yilmaz has been in stunning form this season, scoring 22 goals in 18 Super Lig matches.
That figure is even more remarkable given the forward has contributed a massive 69 percent of the total league goals Trabzonspor have scored this season, with the Karadeniz Fırtınası currently sixth in the table having netted just 32 goals.
Also, his next best goalscoring competitor in Turkey's top-flight this term is Herve Tum of Genclerbirligi, who has nine goals, with Johan Elmander of Galatasaray just behind him with eight goals.
His form is reminiscent of Luis Suarez at Liverpool, carrying on his back a woefully-poor team in attack.
Yilmaz's goals this season—which have led him to be on course to break the Trabzonspor record of 31 goals in a season held by Fatih Tekke—follow on from his 20-goal league haul last campaign.
And his league form has helped him cement his place in the Turkey national team, as he proves himself as a key figure for boss Guus Hiddink.
Burak is one of Europe's best strikers, I'm proud of his recent achievements.
It's clear to see the Turkey striker is a player in phenomenal form at the moment and would clearly provide the competition to Robin Van Persie that Arsenal need.
The only thing is, despite Yilmaz talking up a move to the Emirates, Arsene Wenger's side haven't actually been linked with the forward.
So whether the player or his agent are trying to force through a deal or there is actually interest from Arsenal, it's hard to say.
The Gunners' fans, it seems, would like to see the player come, or at least that's what Turkish media are insisting.
Either way, Yilmaz is clearly a player in the Arsenal mold, thriving in a team that hits opponents fast on the break and have great link-up play—two things the Gunners have statistically done better than any other team in England's top-flight this season.
The Turkish forward is also a footballer well-suited to the Premier League, possessing the right mix of pace, shoulder strength and great aerial ability—perhaps helped by the fact he weighs 88kg and is 188cm tall.
With great finishing and great form on his side, Yilmaz would surely be a better option for Arsenal than Marouane Chamakh and Park Chu-Young.
He's yet to be properly linked with Arsenal, but given that there's public interest from the player himself and the Gunners faithful, could Burak Yilmaz be Arsene Wenger's new transfer target?