Everton are currently at a pre-season training camp in Switzerland and while Moyes is reluctant to part with the player he signed for £8.6m from Crystal Palace in 2006, Johnson could be sacrificed in a bid to kick-start the incoming transfer activity at Goodison Park.
Moyes is an admirer of Blackburn's David Bentley but Tottenham remain favourites to land the England star, while Everton have also been linked with a move for Manchester United midfield man Darren Fletcher.
The Everton boss had previously told the official club website: "Andy Johnson is not for sale - it is that simple.
"He is a highly-valued member of my senior squad and I look forward to seeing him when the players report back for pre-season training."
Johnson has scored 22 goals in all competitions since his move from Selhurst Park.
He only signed a new five-year deal with the club in November 2007 after being linked to the Hammers.
