Liverpool boss Jurgen Klopp says he "could not be less relaxed" about the injury that has ruled Adam Lallana out for a month.
Lallana suffered a thigh injury against Lithuania last Sunday, a game in which he played 90 minutes, just four days after a 66-minute stint versus Germany in Dortmund.
The back-to-back international matches followed a high-intensity 1-1 draw at Manchester City and Klopp felt England's use of Lallana contributed to his setback.
Speaking ahead of Saturday's Merseyside derby clash with Everton at Anfield, Klopp said: "Obviously I was not happy. Somebody said I was relaxed about it, I could not be less relaxed about this - but it is not about blaming Southgate.
"I was not happy he played on the Wednesday but it was not my decision. I respect decisions of other managers.
"Adam loves football so much he didn't say that he should wait another day.
"You can not only play on Wednesday. You have to train again."
Klopp is hoping there will be more dialogue with Southgate and other international coaches in future, adding: "It's absolutely normal to have contact with the national coach. We are in close contact. It's normal. We both share players.
"We are not the only team that suffered.
"No club coaches are happy. I really think we could handle these situations better if we work together. I was never part of the meeting [for fixtures]. As long as we don't work together we will have these situations all the time."
Liverpool have injury worries elsewhere as well with captain Jordan Henderson out after suffering a delay in his return from a foot injury while Daniel Sturridge remains sidelined.
Klopp said of Henderson: "He's had a little setback, I can't say how long it will be.
"It's just a question of time, unfortunately. There's nothing we can do.
"We have to be careful and not take a risk. It's always too long with injuries."
Brazilians Roberto Firmino and Philippe Coutinho returned from international duty unscathed but Sturridge remains a concern.
"No chance [of playing on Saturday]," said Klopp when asked of the England striker's availability.
"I heard he was back in training but I don't know who gave this information. He is working with the rehab coach. No chance this weekend."