The Hammers are in pole position to sign the 16-year-old after he spent time at the Boleyn Ground.
However, he is also attracting interest from a number of high-profile clubs including Arsenal and Juventus.
Nushi is a pacey forward who currently plies his trade in his native Sweden for Stockholm-based club AFC United.
The club's sporting director Hamid Kavodvand has admitted that he expects the player from Albanian descent to make a move overseas in this window.
He told skysports.com: "Rinor is attracting a lot of interest and it is unlikely that we will be able to keep him here.
"At the moment he prefers West Ham after enjoying his time there, but no deal has been agreed at this stage."