The personal terms of the deal are being discussed by the parties.
The Italian forward scored 16 goals in 36 appearances for the Italian side last season and will be joining the Hammers.
There is still dialogue between the clubs and West Ham is interested in a player from the other club.
The 20-year-old is rumored to be leaving the club this summer.
The Irons are waiting for a response from Lille regarding the possible transfer of Amadou Onana.
West ham missed out on signing Jesse Lingard after he joined Forest on a one year deal
The 29-year-old decided to join the newly-promoted side after West ham made a proposal to sign him.
West Ham have made three signings so far this summer with the arrivals of defender Nayef Aguerd, who joined from Rennes for £30m, and goalkeeper Alphonse Areola, who joined from Paris Saint-Germain for $8.25m.
West ham is looking to improve their squad ahead of a season that will see them compete in the Europa League
Kostic is a left-sided player who is entering the final year of his contract at the German side.
The left-back position has been a priority for the club this summer and they have moved for a Serbian player.
Kostic has the ability to play as a left wing-back or as a left flanker, and he had a close look at him last season.
Kostic had the most assists in last season's tournament with six, and he won the player of the season award.
It's easy to see why Gianluca scamacca is in high demand. He scored 16 times in Serie A last season and is capable of many more. He took the route to the top.
Scamacca is both an early and late bloomer. When he was a teenager, he traveled to the Netherlands twice in order to find an opportunity. At the age of 21, he was out on loan at Ascoli.
He could command a fee in excess of £40m if he were to follow in the footsteps of Zlatan Ibrahimovic and become Italy's next top scorer. A man is dancing.
Here, you can read Adam Bate's full feature.
Areas which are often overlooked by the public are the key strengths of Broja. He runs with the ball at his feet when he regains possession in the last third.
The Albanian started only 21 league games last season, but only five players recovered more possession in attacking areas.
He is one of the league's elite players for pressing from the front, as well as being one of the top 10 for fast breaks.
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The 20-year-old was able to create space and carry the ball into dangerous areas despite his limited game time.
The youngster has scored 20 goals in just 48 starts for the club, an impressive strike-rate, given he has also been hooked off 35 times.
Here you can read Adam Smith's full feature.
The transfer window will close at 11pm on September 1st.
Sky Sports has a dedicated Transfer Centre website where you can keep up-to-date with all the latest news. Sky Sports News gives you the ins, outs and analysis of sports.
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