According to Express and Star, West Ham United are keen on signing the 28-year-old goalkeeper as a replacement for Lukasz Fabianski, who turned 36 on Sunday.
Johnstone has been a key player for the Baggies and will be crucial to their survival hopes this term.
The player has previously been linked with Manchester United, Tottenham Hotspur and Leeds United, but a move to the Hammers may appeal.
The Baggies are keen on holding onto Johnstone, but it will be difficult for them to convince him to stay if they drop down to the Championship.
Johnstone is way too good to play in the second tier and is likely to have quite a few offers from the Premier League at the end of the season.
SL View: Johnstone and West Ham are a good match for each other
The Hammers are going well in the Premier League this season and have the chance of securing qualification to the Champions League.
If they manage to secure European football, they should be able to fend off the competition for players like Johnstone.
He would be a regular starter for the Hammers if he joined the club, and it would be the ideal next move in his career.
As far as the other suitors are concerned, United may not be able to provide him regular game time even if they sold David de Gea. Dean Henderson would probably be ahead of him in the pecking order at Old Trafford.
Meanwhile, Illan Meslier has done well for Leeds, and the youngster is unlikely to be dropped right away if Johnstone moves to Elland Road.
West Ham are probably the only interested club that can offer him a regular starting berth next season.