Anfield chiefs have snapped up a host of players from the Saints in recent seasons including Adam Lallana, Dejan Lovren, Nathaniel Clyne and Sadio Mane.
A new midfielder could be a priority for Klopp this summer with Jordan Henderson sidelined through injury, Lucas facing an uncertain future and Emre Can's contract talks dragging on.
Romeu has impressed for Southampton since joining from Chelsea in 2015.
And Thompson believes he would add some much-needed bite to Liverpool's engine room.
He told Sky Sports: "There might be a bit of interest as well in Romeu, who has had an excellent season and does exactly what it says on the tin.
"He's a destroyer, he's got a great attitude and passes the ball very well.
"If Southampton were interested in cashing in then he'd offer us something we don't really have at the moment."
Romeu is under contract at St Mary's until 2021.
He has made 42 appearances in all competitions this term.
