The 24-year-old underwent a medical with the Premier League leaders on Wednesday after they triggered his £7.25m release clause.
Minamino impressed against Liverpool in the Champions League this season, scoring at Anfield as Salzburg finished third behind the Reds and Napoli to qualify for the Europa League.
He has also registered five goals and six assists in the Austrian Bundesliga this season for league leaders Salzburg.
Klopp was asked about the situation after beating Monterrey – before the deal was finalised – and accidentally all but confirmed the deal.
“Is it official yet?” he asked the club’s press officer. “No. Oh, I can’t speak about it.
“But if he could come… he’d have been really good here today, I could have brought him on.
“But we have to wait for that one. Sorry.”