Leading newspaper Marca said on their front cover the 48-year-old has been chosen by Spanish Football Federation president Luis Rubiales as the successor to Julen Lopetegui, who was sacked the day before the World Cup.
The story was echoed by radio station Onda Cero and the website of daily AS.
The Federation have scheduled a media conference on Monday to announce changes to the organisation's structure after Fernando Hierro, who took charge of the team during the World Cup, left his role as sporting director.
The 2010 champions were knocked out of the tournament by hosts Russia in the second round, losing 4-3 on penalties after a 1-1 draw.
A former midfielder for Sporting Gijon, Real Madrid and Barcelona, Luis Enrique won two Liga titles, one Champions League and three King's Cups as Barca coach between 2014 and 2017.
He also coached Celta Vigo and Barca's reserve team and has spent the past year on sabbatical since leaving the Nou Camp in May 2017.
The Spanish Football Federation were not immediately available for comment.
