Henderson arrived at Anfield from Sunderland in 2011 and has become one of the their most important players as club captain.
The 31-year-old is under contract until 2023 but a report from The Athletic suggests initial renewal talks have got off to a poor start.
Sporting director Michael Edwards also doesn't see Henderson's renewal as a priority.
It's a situation that is said to be 'complicated and sensitive' right now as the Reds look to renew other players first.
Goalkeeper Alisson is close to agreeing a new deal, while Fabinho, Virgil van Dijk, Mohamed Salah and Sadio Mane are all players the club want to tie down.
Liverpool are wary of letting a situation similar to former midfielder Gini Wijaldum develop.
Wijnaldum left Liverpool for Paris Saint-Germain this summer after being left disappointed at his proposed wages and contract length.
The Dutch international is 31, the same age as Henderson, who will be 33 when his current deal expires.
Fenway Sports Group are said to be focused on younger players and don't want to table big deals for older stars.
PSG and Atletico Madrid are two clubs said to be monitoring Henderson's status.
Mauricio Pochettino is a fan, while former teammate Luis Suarez could play a role in Henderson moving to La Liga.
Henderson spent this summer at the Euro 2020 with England and helped his country reach the final.
Along the way, he managed to score his first goal in 4-0 victory against Ukraine.
He said: "It's been a long time coming. I felt one was coming. I thankfully managed to get on the scoresheet.
"I'm delighted for the lads, the effort tonight was fantastic and will give us good momentum going into the next game. I tried to counter the best I can.
"On the pitch, I don't need to score more goals and there's other players to do that. I need to concentrate on my job. I would like to have scored more goals over the last few years. I managed to get one today which is always nice."
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