The 46-year-old has led Barcelona to the Champions League title, successive La Liga titles and two Copa del Rey victories since taking charge of the Catalan outfit in 2014.
The Spanish side are on the brink of elimination from this season's Champions League, however, after being thumped 4-0 at Paris Saint-Germain in the first leg of their last-16 tie on Tuesday night.
According to Spanish radio station COPE: "The key players are disillusioned with him. They see him incapable of lifting the team and finding solutions, they do not believe in him."
Enrique reportedly had to be restrained by three people when he confronted a Spanish journalist in the aftermath of the team's heavy defeat in Paris.
The Spaniard's current deal at Barcelona will expire at the end of the season.