According to the Daily Mail, the Spain head coach is among the contenders for the position with 'many admirers at Old Trafford'.
However he's still thought to be behind Erik ten Hag and Mauricio Pochettino in the shortlist.
While Enrique is regarded as an attractive option from a financial perspective, the fact that Spain are due to compete in the 2022 World Cup in Qatar this winter could prove to be problematic.
As well as enjoying a decorated career as a player with Real Madrid and Barcelona, the 51-year-old has an impressive CV in management too.
Taking his first role at Barcelona B, the Spaniard has led Roma, Celta Vigo and Barcelona, where he spent three years and won two LaLiga titles and the Champions League, before taking charge of the Spain national team in the summer of 2018.
Reports last year even suggested that Cristiano Ronaldo was a huge fan of Enrique, wanting the Spaniard to replace then-boss Ole Gunnar Solskjaer.
The man himself wasn't buying the speculation, responding to Spanish TV channel La Sexta, saying: "Is today April Fools' Day?"
While Ronaldo might be a fan of Enrique, Darren Bent believes that United must still prioritise a move for Chelsea boss Thomas Tuchel, should he become available this summer.
"Tuchel is in a different bracket to Ten Hag and Pochettino," Bent told talkSPORT Breakfast.
"Tuchel went to Paris Saint-Germain, got them to the Champions League final, won the league… if you ask me, Tuchel is a top class manager.
"If there was an inkling that Manchester United could get Thomas Tuchel, I'd go all out to get him."