The Daily Mail says in a row that will increase tension between the pair and could hasten Alonso's Anfield departure, the Liverpool midfielder insisted on staying behind instead of boarding the flight for Tuesday's Champions League tie with Inter Milan.
It is understood that he was convinced he would be allowed to link up with his team-mates once partner Nagore had given birth on Merseyside. But Benitez announced that Alonso would not be travelling.
He added: "He wanted to come out here but only if everything was okay.
"We could not wait and had to make a decision. You cannot wait and wait and wait for any player. There comes a time when you have to decide and make use of your squad."