The Gunners are understood to be looking to delve into the transfer market in the New Year, with solving issues at centre-back and defensive midfield on the agenda.
Wenger is said to be worried by the series of injuries suffered in his squad already this season, which have exposed slight lacks of depth in the aforementioned areas of the pitch.
And with Mikel Arteta now edging towards the end of his career, McCarthy has emerged as the man the Frenchman wants to anchor his engine room.
The Republic of Ireland international only joined Everton from Wigan last year in a deal worth around £13m, but his improvement has led to talk of big moves throughout 2014.
Arsenal opening offer of around £16m is unlikely to be looking on too kindly by the Toffees, but the Londoners may opt to go higher in a bid to land the enforcer.
McCarthy is currently out of action with a hamstring injury, but is expected to return to action in around three weeks’ time.