The German boss admitted last weekend that he tried to re-sign the Japan international, who moved to Old Trafford from Dortmund last summer.
"We still keep in touch," Klopp told Bundesliga.de. "It's a bit difficult to speak to him directly, largely because his German hasn't dramatically improved while he's been in England.
"Talking to him on the phone isn't what you'd call a piece of cake. I often have a few words with Shinji's agent, Thomas Kroth, as well. Believe me, I'm always right in the picture about what the lad's up to."
Kagawa previously admitted that he was disappointed with his first season at Old Trafford, which ended in the club winning the Premier League title.