The 25-year-old has been on loan at Aston Villa since January 2017, helping to guide the Villans to the Championship play-off final this season - eventually suffering a 1-0 defeat to Fulham at Wembley last month.
Johnstone would have likely moved to Villa Park on a permanent basis had they been promoted, but the former England Under-20 international is unlikely to be swayed by the prospect of another season in the Championship.
The Guardian have now outlined how West Ham are stepping up their pursuit to sign a new goalkeeper this summer, with Johnstone identified as a candidate to replace the departed Joe Hart.
The Hammers currently only have Adrián in their senior goalkeeping ranks, with the 31-year-old expected to reclaim the number one jersey next season after impressing at the back end of last season. He has been linked with a move back to La Liga however, as he seeks to establish as a regular first-choice goalkeeper.
Johnstone is out of contract at Old Trafford next month, though United do have the option to extend his deal by another year, a measure which has all but been confirmed as the club.
He is unlikely to be keen on warming the West Ham bench next season, but may relish the opportunity to link up with new manager Manuel Pellegrini, with the Chilean having recently taken over at the London Stadium.