Moyes has insisted Johnson, 27, is not for sale - but Everton may now decide to cash in to boost their transfer fund if they receive offers of about £10m.
BBC Sport understands Wigan boss Steve Bruce is currently in pole position and is ready to offer Johnson a lucrative package to tempt him to the JJB.
But Sunderland counterpart Roy Keane is still hoping to clinch a deal.
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 and Manchester United midfield man Darren Fletcher.
Ameobi's career at Newcastle United is said to be over, and West Brom are still interested in taking the out-of-favour striker.
The Baggies are reported to be preparing a £2.5m bid for the former England Under-21 man but only IF former Sunderland striker Kevin Phillips joins Birmingham City, as expected.
The Baggies are reported to be preparing a £2.5m bid for the former England Under-21 man but only IF former Sunderland striker Kevin Phillips joins Birmingham City, as expected.