Dwyer bags five goals in 299th kookaburras appearance.
The 23-year-old midfielder has impressed since joining from Fulham a year ago and has scored four goals and assisted another in his first 16 Bundesliga appearances.
Moody’s decision could see the former Scotland international slip down the pecking order in next month’s opening day squad, though he made his first-team debut in last Sunday’s 5-1 win over Schalke.
He has scored 12 times in 49 appearances since leaving Fulham before the 2015-16 season but remains one of Moyes’ most frustrating players.
The 26-year-old had a year remaining on his loan deal at Goodison Park but left after scoring three times in six games in the league last season.
“I have an ambitious task ahead of me which is to create something in this gro이천안마up which is strong enough to reach the big occasion,” he said.
“It was important, after I had spoken to all the players individually, to make sure that we could improve. We have a lot of quality in the squad so if we can improve it all the more we will improve.”
Meanwhile, Everton are expected to meet their winter targets – including a switch for midfielder Kolo Touré – by next week.
The Blues are also said to be ready to give Ronald Koeman a new contract before he steps down from his current role with Everton this summer.
The Koeman era began in spectacular fashion when he took over from Roberto Martinez in October 2012.
And the Dutchman has continued that run with 11 wins in 20 games, which has put him in the running for the Premier League Golden Boot with five assists, one goal, and the best goal tally in his 13 appearances with the club.
He can expect to continue under Moyes as the Dutchman is keen to reward his first-team squad for an impressive campaign which includes a win at Burnley last weekend.