The Baggies have been chasing a striker ever since the transfer window opened, with Grant the number one target for both head coach Slaven Bilic and technical director Luke Dowling.
With money tight at The Hawthorns, talks with Huddersfield have dragged on while the clubs worked to structure a deal.
But it is believed there has now been a breakthrough and a deal has been agreed, giving Grant the green light to undergo a medical today.
It had been expected the 23-year-old would initially join the Baggies on loan.
The move would then have become permanent if certain clauses were met.
But according to reports, Grant is now set to join the Baggies permanently for a fee in the region of £15million.
That figure will be paid in instalments over the course of his contract, which is likely to be either four or five years.
While Grant looks set to arrive, fellow striker Kenneth Zohore is expected to leave The Hawthorns before the transfer window closes at 5pm tomorrow.
Derby reportedly made a late move to sign the striker after fellow Albion forward Charlie Austin turned down the chance to move to Pride Park.
But the Express & Star understands Sheffield Wednesday are Zohore’s likely destination – with the Owls having been chasing the Danish hitman for the past few weeks.
Meanwhile, midfielder Filip Krovinovic says he’s been impressed with the new players Albion have bought this summer.
The Croatian first joined the Baggies on loan from Benfica last season and rejoined the club in a similar deal last month.
Prior to his return, though, Albion bolstered their ranks by bringing in David Button, Cedric Kipre, Branislav Ivanovic and Conor Gallagher.
And Krovinovic says all four of his new team mates have made a big impression in training.
“English people are unbelievable – Conor (Gallagher) is English and he is a very good guy,” Krovinovic said when asked about Albion’s new recruits.
“He has so much potential, he is only 20 years old, but he is a very good player.
“He is going to help us a lot this season.
“Branislav (Ivanovic) – we speak the same language, with the accent, some words are different. But when you speak the same language it makes it easy.
“We have a very good relationship and his experience and attitude will help us.
“He can speak to us younger players, he will shout at us and sometimes you need that.
“Cedric (Kipre) is a very hard working central defender. (David) Button is a good goalkeeper and also a good guy.
“Technically he likes to play with his feet and I like it when the goalkeeper isn’t scared to play.
“Everything is positive about all the new signings.”