Barkley made an impressive senior debut for the Toffees in Saturday's 1-0 Barclays Premier League defeat to QPR and having kept his place in the starting line-up, put in another eye-catching display on Wednesday evening as Moyes' side came from behind to beat Sheffield United 3-1 in the Carling Cup second round.
"I spoke to Stuart Pearce and he is ready for the Under-21s," Moyes said of the 17-year-old. "But the boy has not played enough games of football. He made a couple of mistakes in the first half with the ball - you will find there will be times when he does things and you think it's not maybe quite right."
He added: "That is because he missed a year and has been plunged right in at the deep end."
Barkley was involved in two of Everton's goals in the clash with the Blades at Goodison Park as the Merseyside outfit recovered from an early scare.
Their npower League One opponents took the lead against the run of play thanks to Richard Cresswell's effort just before the half-hour mark, but the same player diverted the ball into his own net moments later after goalkeeper Steve Simonsen had got a palm to a strike by Jack Rodwell.
The Toffees then took control of proceedings as Victor Anichebe converted Barkley's drive to put them ahead in the 37th minute.
And five minutes later Mikel Arteta increased the advantage, collecting the ball and curling it home when Barkley's surging run had been halted on the edge of the box.
Source: PA
Source: PA