Woodgate, 36, retired at the end of last season after helping hometown club Middlesbrough achieve promotion to the Premier League.
The former England international will scout for the Reds in Spain and Portugal, countries that have been utilised with varying success in recent seasons.
Alberto Moreno, Javier Manquillo, Iago Aspas, Luis Alberto and Aly Cissokho have arrived from Spain while Joao Carlos Teixeira, Tiago Ilori and Lazar Markovic have been signed from Portuguese clubs.
Woodgate signed for Real Madrid from Newcastle in 2004 for £13.4 million, joining David Beckham and Michael Owen at the Bernabeu and playing alongside Zinedine Zidane and Ronaldo.
In 2008 he scored the winning goal as Tottenham beat Chelsea 2-1 to win the League Cup.