Carragher sees Emery as the ideal option should United decide against handing Michael Carrick the job long-term.
"Unai Emery should be heading to Old Trafford this weekend as the front-runner to become the next Manchester United manager," Carragher wrote in his column for the Telegraph.
"I suspect he does not rank high among the candidates. That is good news for Aston Villa, but it must be frustrating for one of the best, and still most underrated, coaches in Europe.
"Emery is one of the few symbols of a bygone era when managers wanted and assumed complete control of a club's football operations.
"Once he was thriving at Villa, it was an easy decision for the hierarchy to allow Emery to proceed with minimal interference.
"Whenever Emery's name is referenced in connection to a 'bigger' job, Villa fans argue that he would never have the same love and power at one of the traditional Champions League teams, and they have a point. Sporting directors are proving adept at making themselves the most important people in the room.
"Appointing Emery is effectively a means of enhancing the role of the manager and diminishing that of the directors. That would therefore be a likely cause of friction in the future."