The 28-year-old goalkeeper is facing an uncertain future at The Hawthorns.
He has a contract at the club until 2022, and West Brom want to sell him now rather than lose him for free next summer.
The player also wants to continue in the Premier League, but the Baggies haven’t received any suitable offers for him.
The Athletic claim West Brom had previously set a tentative asking price of £20 million at the end of last season.
However, they are now ready to lower their demands and offload him for around £12m, which seems a decent value for a quality goalkeeper.
West Ham are the only club to have made an official approach to sign Johnstone but have only offered around £6m.
The Baggies believe that there is still plenty of time remaining in the window for other bidders to come to the table.
Although Albion are still planning for life in the coming season without Johnstone, he will be reinstated as the first-choice goalkeeper if he fails to secure a move away.
West Ham are working on other deals with signing a striker and an attacking midfielder a priority for them.
The Hammers have been trying to sign Jesse Lingard permanently, but they have now turned their attention towards signing Ross Barkley.