The Baggies had until 11pm last night to sign players from rival Premier League clubs and from teams abroad.
But they have until October 16 to make additions from the Football League.
Slaven Bilic has made it clear on numerous occasions he is determined to land a striker before the window shuts.
The Express & Star understands Grant is his number one target.
It is also understood the 23-year-old wants to leave Huddersfield to move to The Hawthorns.
Albion were also interested in signing Senegal international Habib Diallo.
But while they made an enquiry to French side Metz to check on his availability and price – they did not shore up that interest.
Diallo instead completed a deadline day move to Strasbourg.
That means the Baggies now only have the EFL to shop in as they work to bring in a goalscorer.
The club has been locked in talks with Huddersfield over a move for Grant all summer. With money tight at The Hawthorns, Albion want to initially sign the forward on loan and are willing to pay a substantial loan fee.
The deal will then become permanent if the Baggies successfully retain their Premier League status for a second season.
While the clubs still remain apart on how to structure the deal, there is an overall acceptance an agreement is likely to be reached before.
Grant hasn’t featured for Huddersfield yet this season. And that is another reason Albion are eager to conclude the deal as soon as possible.
With no game at the weekend due to the international break, Bilic wants as much time as possible to work with the former Charlton man.
If a deal with Huddersfield cannot be agreed, however, the Baggies will turn their attention to signing one of Andre Gray or Troy Deeney.
Both players have been made available for transfer by Watford.
Charlie Austin is believed to be in talks about a potential move to Derby.
But it is understood the striker would need convincing to move and would rather stay and fight for a place at Albion.