But according to the reports from Turkey - where the 27-year-old plies his trade with Bilic's former team - last week's sacking of Roberto Martinez could put paid to interest from Merseyside.
Co-owners ready to spend
The defensive box-to-box midfielder has a contract until 2019 at Besiktas which highlights how highly the Turkish Super Lig side rate him.
The Hammers look set for another busy summer in the transfer market with their big move to the 60,000 seater Olympic Stadium looming large.
Co-owners David Sullivan and David Gold were true to their word by investing significant funds to strengthen Bilic's squad last summer.
And they look set to go for it again with the club targeting two top strikers to elevate them to the next level.
Hammers need to bolster midfield
Hammers fans will be pleased to hear that Kavlak is very similar to their former cult hero Valon Behrami, now with Watford.
Kavlak is a key man in midfield for club and country with his tireless running and seemingly endless energy levels. He should be able to adapt to the Premier League too as he was one of Besiktas' star players in their matches against Liverpool and Arsenal in Europe last season.
A hardworking leader much like Behrami, Kavlak could became a real fans' favourite at Upton Park.
He is often paired up in central midfield with a more attacking player at the tip of a diamond which would fit well with the likes of Dimitri Payet and Manuel Lanzini.
With Alex Song looking set to return to Barcelona after failing to prove his long-term credentials the Hammers will need to beef up their midfield.
But there is a word of caution in that Kavlak has also been out all season and two months of last season through injury, although he is expected to be fit for Austria come the Euros next month.