The Baggies have been working on a deal to sign the striker for the past few months in what has become a long and drawn-out transfer saga.
Huddersfield initially slapped an £18million price tag on the forward.
But with money tight at The Hawthorns – that fee was beyond Albion in one lump sum.
With the transfer window closing tomorrow, though, the Terriers and Albion have been working on other ways to structure a deal.
A loan that would have seen the Baggies pay a substantial loan fee was discussed – with the move then becoming permanent at a later date.
But now the two clubs have agreed a permanent transfer that will see Albion pay £15million over the next six years – the length of Grant’s contract.
The 23-year-old has been the number one striker target for head coach Slaven Bilic and technical director Luke Dowling all summer.
He is the ninth player the Baggies have signed since winning promotion.
And the Express & Star understands he will be the last player the club brings in this window with no further incomings expected today.
Deadline day, though, could still pass with an outgoing from The Hawthorns.
Sheffield Wednesday are long-term admirers of Kenneth Zohore and are still believed to be trying to sign the forward.
If completed before the 5pm deadline, it is likely the Dane will initially join the Owls on loan with Albion still paying a percentage of his wages.
While Zohore could still depart, Charlie Austin looks set to stay with Albion.
The 31-year-old had interest from both Derby and Cardiff but has opted to remain and fight for his place. Reports yesterday, though, suggested Austin is open to moving to America and the MLS in January.