Spain legend Fernando Hierro guided his team through the group stage but was powerless to stop them being dumped out of the tournament on penalties by hosts Russian Federation in the round of 16.
Earlier on Monday, the federation announced Jose Molina has replaced Hierro as sporting director.
Enrique, 48, won two LaLiga titles and the Champions League with Barca before stepping down a year ago.
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Richard Blumenthal accused Trump of outsourcing his decision to the Federalist Society and other conservative groups. Trump has said he'll announce his selection Monday in a prime-time event from the White House East Room at 9 p.m.
Luis Enrique's appointment shows the ambition of the federation, who are hoping he could put in place a team that is both competent and hungry to make things right going forward.
Luis Enrique, who won nine trophies with the Catalan giants between 2014 and 2017, has been confirmed as new La Furia Roja boss by the Spanish FA.
Fernando Hierro was given the job on a caretaker basis, to at least initially see Spain through this summer's World Cup.
This has resulted in a newfound apathy across the national team fanbase and Luis Enrique must find a way to bring back the trust and optimism of the old days.
Rubiales said: "We know that Luis Enrique is a person of character".
Enrique won the La Liga, Spanish Cup and Champions League treble while in charge of Barca in 2015.