Griffiths, who enjoyed a successful loan at Lincoln last season, has been heavily linked with a temporary switch to Portsmouth.
Albion’s goalkeeping position is currently up for grabs following the departure of Sam Johnstone.
David Button is currently in pole position with Alex Palmer expected to be his understudy.
But manager Bruce is also looking at Griffiths – who has been tipped to have a bright future at The Hawthorns – while the team is away training in Portugal.
Asked about the strength of the goalkeeping position heading into the new season, Bruce said: “I’m very happy with the situation.
“Button is in charge of it at the minute because he had such a good end to (last season).
“And the young one (Griffiths) who has come back from a loan too so I’m very happy in that department. We’ve no problems there.
“With Josh it’s difficult. At this moment in time, do you let him push with the other two, or do you try and find a progression because he’s played over the last couple of years.
“We’ll see what develops because what’s best for him is crucial.”
Meanwhile, the Express & Star understands there is no truth to reports linking Albion with a move for Tom Lawrence. The Wales international is a free agent after leaving Derby.
But despite reports to the contrary, Albion are not believed to interested in signing him – with Bruce’s squad well stocked with players who can play on the right and left wing.
Instead, the boss is understood to be in the market for a striker who can compete with and provide cover for Daryl Dike.
A central midfielder is also a target as is cover at full-back.