Rooney, the former Manchester United and England striker, was appointed Birmingham manager on Wednesday.
The new manager replaces Steven Gerrard, who was sacked last week.
The club announced Gerrard’s exit from the club in a statement shortly after Fulham defeated them 3-0 in the English Premier League game on Thursday.
Sadio Mane’s second-half winner secured Liverpool a vital 2-1 victory at Aston Villa on Tuesday to keep the pressure on Manchester City in the EPL title race.
This move will see Coutinho reunite with his former teammate Steven Gerrard, who is coach at Villa Park.
Gerrard, 41, who joined the club last month, will miss Sunday’s home game against Chelsea and Tuesday’s trip to Leeds United.
