Mirko Slomka's side - who also allowed two players to leave - only avoided relegation from the German top flight in a play-off last season, and the coach has recognised the need to bring in reinforcements after taking one point from their opening two games this time around.
Holtby and Green signed season-loan loan deals at the Imtech Arena on Monday.
Attacking midfielder Holtby has struggled to make an impact at Tottenham following his move from Schalke in January last year.
The 23-year-old was loaned out to Fulham for the second half of the last campaign as the Craven Cottage outfit were relegated from the Premier League.
Bayern Munich winger Green is yet to make his bow for the Bundesliga champions but the 19-year-old has earned three caps for the United States.
The two to leave Hamburg were Per Skjelbred, who joined Hertha Berlin on a three-year contract, and Milan Badelj, who departed for Fiorentina after agreeing a four-year deal.
Mainz also added to their ranks with the loan signing of midfielder Jonas Hofmann from Borussia Dortmund.
Rock-bottom Augsburg terminated the contract of striker Aristide Bance by mutual consent, while Mathias Fetsch left the club on loan.
Elsewhere, Paderborn announced the signing of defender Rafa from Getafe on a two-year contract.