Lopetegui, who has coached the Spanish national team for the past two seasons, will coach Real for the next three seasons, the statement added.
Real Madrid winger Gareth Bale is a long time Manchester United transfer target, so today's headline news emerging from the Santiago Bernabeu has naturally invited more speculation about his future. And as they had hoped, the European champions have announced the Frenchman's successor ahead of this summer's World Cup.
And United supporters are concerned that new Madrid boss Lopetegui will use Spain's World Cup campaign as a chance to lure De Gea away from Old Trafford.
The Spanish giants have been on the hunt for a new manager ever since Zidane surprisingly stepped down last month, just days after winning his third consecutive Champions League trophy.
This marks a return to club football for the Asteasu, Spain native, who took over La Roja following the retirement of Vicente del Bosque after Euro 2016.
He also played for both Real Madrid and Barcelona, although he only played one league game for Madrid, and five for Barcelona.
He previously had a spell as manager of Porto, as well as with the Spanish national youth team set-up and at Real's B team.
However, they have now made a decision to appoint Lopetegui, who is currently concentrating on managing Spain in Russian Federation.
It remains to be seen if he will head to Real as a World Cup victor, though Spain's squad is stocked with exciting talent.