The Baggies have been on the hunt for a new centre forward ever since the transfer window opened.
Grant is understood to be the primary target for both head coach Slaven Bilic and technical director Luke Dowling.
But despite the player wanting to make the move, negotiations between the two clubs have dragged on.
Now, though, it is understood the clubs are close to agreeing terms as all parties work to complete the deal before the transfer window closes on Friday.
It is believed the two clubs are close to finalising a deal that will see Grant initially move to The Hawthorns on loan.
If certain clauses are met, Albion will then be obligated to buy the player at the end of the season.
There is an acceptance from all parties that a deal is likely to be reached before Friday’s 5pm deadline.
But should it fall through, Albion have lined up Troy Deeney and Andre Gray as alternatives.
Both players have been made available for transfer by Watford.
While Grant is likely to arrive at The Hawthorns, fellow striker Kenneth Zohore is close to leaving the club, the Express & Star understands.
Albion are in advanced talks with Sheffield Wednesday as they look to offload the 26-year-old.
And it is understood a deal that will initially see Zohore join the Owls on loan is likely to go through before the window closes.
One player who looks set to stay with Albion, though, is Charlie Austin.
According to reports, the 31-year-old has turned down the chance to join Derby.
Cardiff are also thought to be interested in the former Southampton man who the Baggies are willing to offload.
But it is believed the striker wants to stay and fight for his place.