Can’s later return is due to his international commitments with Germany at the Confederations Cup, which only ended a week ago.
The 23-year-old midfielder will, therefore, have another two weeks away, with the German joining the squad during the training camp in his homeland later in July.
Lallana and Coutinho are later returnees due to the international commitments with England and Brazil, respectively, in June.
It marks a change from last summer when the Euros meant many players returned to the Reds’ pre-season much later.
Most of those who are back in training will, therefore, take part in the friendlies against Tranmere (Wednesday) and Wigan (Friday), with Klopp likely to play two different XIs in each fixture this week.
Players will then start playing more than 45 minutes next week in the two Asia Trophy games in Hong Kong, before heading to Germany where the Reds will be based 30 miles outside of Munich.
Meanwhile, Klopp will be hoping that a couple of new signings can join the squad by then. The last arrival last summer was Gini Wijnaldum, whose signing was completed on July 22nd.
Key dates in Liverpool FC’s pre-season diary this summer:
July 12: Friendly: vs. Tranmere
July 14: Friendly: vs. Wigan
July 19: Asia Trophy: vs. Crystal Palace (Hong Kong)
July 22: Asia Trophy: vs. Leicester or West Brom (Hong Kong)
July 29: Friendly: vs. Hertha Berlin (Berlin)
August 1: Audi Cup: vs. Bayern Munich (Munich)
August 2: Audi Cup: vs. Atletico or Napoli (Munich)
August 4: Champions League playoff qualification draw
August 5: Friendly: vs. Athletic Bilbao (Dublin)
August 12: New Season: Watford (A)
August 15/16: Champions League playoff first-leg
August 19: First Home Game: vs. Crystal Palace (H)
August 22/23: Champions League playoff second-leg