David Moyes has enjoyed a productive summer in the transfer market, bringing in the likes of Nayef Aguerd and Flynn Downes from Rennes and Swansea City respectively.
However, the Scot remains eager to add to his creative options with a replacement required for Andriy Yarmolenko and competition needed for Michail Antonio.
According to transfer expert Fabrizio Romano, the Hammers have launch a bid to sign versatile wide player Kostic.
The Serbia international has spent the last four years at Frankfurt, contributing 33 goals and 63 assists from 169 appearances in all competitions.
The 29-year-old played a key role in Frankfurt's Europa League success during 2021-22, chipping in with three strikes and six assists.
While Kostic now has the opportunity to play Champions League football for the first time in his career, it appears that he will be left with a choice to make regarding a switch to the Premier League.
Despite his status as a key man at Frankfurt, Kostic has less than 12 months remaining on his contract, realistically leaving him available for transfer for the right price.
Romano suggests that conversations have already taken place with the left-sided player having the same agent as fellow West Ham target Gianluca Scamacca.
Although West Ham's interest in Chelsea's Armando Broja has been well documented, it has been suggested that a compromise is more likely for Scamacca.
The East Londoners are said to be waiting to strike an agreement with Sassuolo with a different report saying that an offer in excess of £34m is on the table.
The Italy international had been attracting attention from Paris Saint-Germain before the Ligue 1 champions instead opted to sign Hugo Ekitike.
Scamacca, who is 23 years of age, netted 16 times from 36 outings in Serie A last season.