Bilic has already signed Norweigan Havard Nordtveit and Algerian star Sofiane Feghouli on shrewd free transfers this summer with the likes of James Tomkins, Alex Song, Victor Moses and Emmanuel Emenike all departed.
Diafra Sakho and Enner Valencia look set to follow them out of the exit door as Bilic reshapes his squad for a busy season which will include Europa League football should the Hammers make it through qualifying.
Tore in action for Besiktas
Signing has not been straightforward
The signing of Tore has not been straightforward, though.
It is understood the club, on the verge of their big move to the 60,000 seater Olympic Stadium, will have to spend as much as £13 million to get their man and perfer a loan with an option to buy.
Frustrated Slaven Bilic
Hammers made to wait
There has been a lot of excitement among Hammers fans over Tore's potential arrival and an announcement was expected last week.
It never came, though, despite pictures emerging online of Tore at Upton Park - where the club's administrative functions are still based until the Olympic Stadium is ready.
West Ham manager Slaven Bilic celebrates
Slav's nearly got his man
But now Sullivan has moved to reassure supporters that the deal is not dead by clarifying the situation in a post on Twitter.
"Update: Tore is almost a West Ham player, should be done early next week," Sullivan said on the social networking site.
"Just a few minor things to sort out!"
Update- Tore is almost a West Ham player, should be done early next week. Just a few minor things to sort out!— Jack Sullivan (@jsullivanwhu) July 8, 2016