The Merseyside outfit were 7-0 winners against Maribor on Tuesday night to secure their first Champions League group stage win in Slovenia at the third time of asking.
Liverpool vice captain Milner started the European clash as Klopp opted to rest skipper Jordan Henderson ahead of this weekend’s trip to north London in the Premier League.
Milner produced an assured performance in the heart of the Liverpool midfield, although the retired England international failed to get on the score sheet.
Former Liverpool defender Thompson believes Klopp has some thinking to do ahead of their top-six clash against Spurs at White Hart Lane on Sunday afternoon.
“I think Milner has given Klopp a massive headache,” Thompson told Sky Sports.
“I know it was against Maribor but he created a lot from the middle in a more offensive role.
“He was high energy and started a lot of the pressing and I actually thought he was the man of the match.
“It’s obviously up to Klopp to determine whether that performance against the opponent in question warrants a start, but his energy on a big pitch like Wembley is going to be important.
“It could be at the expense of Henderson because Emre Can was the holding midfielder player, so that’s probably the only decision to make.
“He was rested Henderson, which was required after he played 180 minutes in playing both games for England, including on an artificial pitch, and Milner certainly took his chance.”
Liverpool finished in fourth place in the Premier League table last term to secure a return to the Champions League for the first time since 2014-15.
Klopp’s men played out a goalless draw with bitter rivals Manchester United in the Premier League last weekend as the Red Devils managed to frustrate Liverpool.
Milner moved to Liverpool from Manchester City in a free transfer in the 2015 summer transfer window.
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