"Divock returns to the Premier League to further add strength to our squad. We wanted to bolster our attacking options and Divock certainly helps us to do that," Forest Chief Football Officer Ross Wilson said in a statement.
Origi joined Milan on loan from Liverpool last summer after making 36 appearances in all competitions, including eight in the Champions League.
The 28-year-old joined the Merseyside club in 2014 and made 175 appearances, scoring 41 goals and providing 18 assists.
During his eight-year spell he won an FA Cup, League Cup and Premier League title with Jurgen Klopp's side, as well as scoring in semi-finals and final of Liverpool's 2019 UEFA Champions League-winning campaign.
Vlachodimos, who was ever-present last season as Benfica won the Portuguese title, has signed a four-year deal.
He will bring a wealth of experience to the City Ground having played 225 times for Benfica in all competitions.
The German-born keeper kept 21 clean sheets in 34 league appearances last season.
"He is very talented and brings an abundance of experience with him, and he will add real competition to our goalkeeping department," Wilson said.