Lovren was part of the squad as the Reds beat Hertha BSC in Berlin on Saturday but was not used in the friendly as a precautionary measure due to a muscle issue.
The centre-back will be available to feature when Jürgen Klopp’s side face Bayern Munich and either Atletico Madrid or Napoli, however.
“He trained and looks fine so far,” the manager told Liverpoolfc.com. “He will be considered for selection in the two games if there is no reaction.”
Karius and Milner have also trained ahead of the tournament at Allianz Arena and come into consideration for selection, but Nathaniel Clyne (back) and Danny Ward (back) remain sidelined.
Having been rested for the 3-0 victory over Hertha at the weekend, Trent Alexander-Arnold, Marko Grujic and Ben Woodburn are fit and in contention for the games on Tuesday and Wednesday.
The boss also explained the thought process regarding Danny Ings’ recovery and why it is right for Liverpool to remain cautious and patient with the striker.
“He is in a really good way and we are very pleased with what he is doing,” said Klopp.
“But it is smart and responsible for him that we allow him maximum time and sessions – both individually and with the team.
“So maybe not involved in the matches just yet but that means nothing more than we take our time and make him strong and ready to be a part of what we are going to do.”