A host of clubs in League One are believed to be chasing the shot-stopper, who impressed during a temporary spell with Cheltenham last season.
And it is understood Albion have given the green light for the 19-year-old to go out on loan again next year.
Griffiths kept 21 clean sheets in 44 appearances to help the Robins to the League Two title.
His impressive displays also saw him earn a first call-up to the England under-21 squad.
That form has led to interest from a host of clubs.
Sheffield Wednesday, Portsmouth and Lincoln are all understood to want to sign Griffiths for next season.
Cheltenham chairman Andy Wilcox, meanwhile, has confirmed his club’s want to re-sign the 19-year-old.
Wilcox has also claimed Griffiths wants to return to Whaddon Road.
He said: “Ideally, we’d want Josh Griffiths back on loan as number one, but we’ll see how it goes.
“Firstly, he wants to come back to Cheltenham, which is the most important thing. He really enjoyed it here.
“I watched him from game one to his last of the season and he’s improved no end.
“He’s just got better. He has a big future, that lad.”
Despite thoroughly enjoying his time at Cheltenham last season, it is believed Albion are more inclined to send Griffiths to a side that is likely to be competing at the top end of League One next season.
Sheffield Wednesday, Portsmouth and Lincoln are all expected to be battling it out for promotion.
And the Baggies have strong ties to Sheffield Wednesday and Lincoln in particular.
Former boss Darren Moore is in charge of Wednesday, while Alex Palmer, Callum Morton and Jamie Soule all had loan spells with the Imps last season.
Griffiths is very highly rated at The Hawthorns and has long been tipped as a potential first-team player.
But it is understood the Baggies are keen for him to go out and play regularly next season, despite Albion’s number one jersey potential being up for grabs.
Sam Johnstone shone in goal for the club last season.
But his impressive performances have led to speculation he will stay in the Premier League – with West Ham believed to interested in the England international.
The Baggies also have David Button and Alex Palmer on their books – with those two hoping to get a chance next season if Johnstone does leave.
Alongside Griffiths, Ted Cann is another teenage goalkeeper in Albion’s youth setup that the club have high hopes for.
At the end of last season, the Baggies activated an option to extend his contract with the club for a further year.