The Bees, along with Villa and several unnamed clubs, continue to chase the 22-year-old after having a club-record £30m bid rejected last month. They first had an offer turned down for him in January 2022.
Forest insist Johnson, who is under contract until 2026, remains part of their plans but his suitors believe they can drive down the asking price to £40m.
The Bees have several attacking options, including Kevin Schade, bought for a club-record fee of £20million last month, their previous club-record signing Keane Lewis-Potter, plus Bryan Mbeumo and Yoane Wissa.
Brentford have also signed goalkeeper Mark Flekken from Freiburg as David Raya faces an uncertain future.
Any sale will be unpopular at the City Ground but Forest will be tempted to balance the books after spending £150m on 27 players last season, putting them among the top five spenders in world football.
