The new Anfield boss believes that the Iceland international had the chance to join a real "football institution" but instead chose Spurs for reasons unknown to him
The Iceland international enjoyed an impressive loan spell at Swansea City last season, scoring seven goals in 19 appearances and the switch looked set to become permanent in May with a £6.8 million fee agreed between Hoffenheim and the Welsh outfit.
However, Rodgers' departure from the Liberty Stadium to join the Reds led to the deal collapsing and, though linked with a move to join his former boss, the 22-year-old instead signed for Tottenham, becoming Andre Villas-Boas' first signing at White Hart Lane.
The Liverpool chief feels that it may prove to be a mistake, though, telling his club's official website: "I brought him to Swansea and he did very well for me there, scoring seven goals in 19 games, and was very good.
|SIGURDSSON JOINS SPURS
|Iceland international joins Villas-Boas' north Londoners
"I then became the Liverpool manager and that then wasn't something that was going to happen at Swansea so he then had a choice of where he wanted to go.
"I knew what the market was and I wasn't prepared to pay anything over what I had known was agreed before.
"Liverpool would have provided Gylfi with a wonderful opportunity to perform with a manager that he knows and at a club which is a real footballing institution. But he's decided to go to Tottenham, for whatever reason.
"I wish him the best; he's a good kid and there's no ill feeling. We've got other targets and we'll move on.”
Despite missing out on Sigurdsson, the former Reading and Watford boss maintained that work is ongoing behind the scenes to make significant additions to what he believes to be an already-very-talented squad.
"I've got a bunch of wonderful players here," he insisted.
"I'm going to take time to have a look at that. I believe there's one or two areas that I'll need to improve on for sure but overall the core of the group is strong.
"Like any manager, you want to add the other pieces that can complete your jigsaw and that's something that we're doing over a period of time.
"We're doing a lot of work off the field in terms of potential players to come in, but it will only be a certain type of player that will fit the model here at Liverpool."