West Brom are expected to make a striker signing this week.
But the Baggies are not pursuing a move for Bournemouth's Josh King who is believed to be seeking £90,000 per week – what the Cherries paid him in the Premier League.
It is believed the Terriers are asking for between £16million and £18million for Grant who scored 19 goals for them last season despite their struggles in the Championship.
With money tight at The Hawthorns, Albion are hoping to negotiate a deal that would see them initially sign the 23-year-old on loan.
The deal would then give Albion the option to purchase the player for a set-price at any time, or they would be obligated to buy him if they retain their Premier League status for a second year.
Head coach Slaven Bilic and sporting director Luke Dowling both see Grant as the final piece of the jigsaw for Albion this window.