But the Rangers boss has played down these suggestions, insisting he is already managing at a “top club”.
Rangers are currently level with arch-rivals Celtic at the top of the Scottish Premiership.
They have won seven of their last eight games, with Gerrard continuing to implement a strong attacking style of play at Ibrox.
His successes have earned him plaudits throughout the footballing world, with fans believing he will be Klopp’s long-term replacement at Liverpool.
The Reds legend was the academy coach at Melwood before moving to Rangers, and it seems inevitable to many that he will return one day.
Klopp, having won the Champions League and taken the Reds to top of the table within the last few months, doesn’t look ready to leave any time yet, and that seems to be fine with Gerrard, who played down the links.
“I am very driven and ambitious and I haven’t gone into this management lightly,” the Liverpudlian said.
“I want to be the best I can be and I feel like I have got fantastic support staff with Mick Beale, Gary McAllister, Tom Culshaw and Jordan Milsom.
“We feel like we have got a really good team and we are enjoying it.
“I get this question a lot about where I want to end up and what is the ambition and the dream – the dream for me is to just win the next football match.
“That is all I need to focus on.
“I don’t need to put time-frames on this or that.
“I am really enjoying what I am doing and I am privileged to be in the position and the focus is the next three points or the next challenge. That is all I worry about.
“It will take me wherever it takes me, if that is to the top fantastic but I already feel like I am working for a top club now.”