The Premier League side have sent scouts to watch the 25-year-old recently.
The Reds had scouts in attendance for the Bundesliga side's draw with Wolfsburg at the end of last month, The Sun claims, and now Jurgen Klopp is seemingly considering whether to bring him to Anfield at the end of the season.
However, The Sun suggests that Cologne would want around £10m for the defender, who has arguably been their strongest player this term.
Should Hector move to Merseyside, he will provide competition for Alberto Moreno. The Spaniard is unquestionably exciting going forward, but defensively, he has often been guilty of mistakes. In particular, the penalty he gave away during the dramatic victory over Norwich in January highlighted how he still has plenty of work to do before he can be considered a top defender.
As well as playing on the left-hand side of defence, Hector brings a versatility that would undoubtedly make him a major asset to Klopp's squad. He has also been used in both a holding and central midfield role this season, which may help persuade Liverpool to meet his asking price if they are interested in a deal.