As for Johnson, the former Birmingham and Crystal Palace hitman is thought to be keen on a move back to London for family reasons.
He joined Everton from Palace for £8.6m in May 2006 and has scored 22 goals in 74 appearances for the Goodison Park outfit.
Everton have yet to make a signing this summer and Moyes has conceded he may need as many as five or six new players.
Moyes has been linked with Moutinho, plus Tottenham striker Darren Bent and Real Zaragoza's Argentine forward Diego Milito.
In sharp contract to Everton, Fulham have gone on a spending spree since securing their Premier League status at the end of last season.
They have already swooped to sign Mark Schwarzer, Andranik Teymourian, Zoltan Gera, David Stockdale, Toni Kallio, Bobby Zamora and John Pantsil and are poised to snap up Swedish defender Fredrik Stoor too.
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