Brazilian playmaker Coutinho, who joined Liverpool in 2013 for £8.5m, was the subject of multiple bids from Barcelona last summer.
He even handed in an official transfer request in a bid to force through the move, but Liverpool remained adamant that he would not be sold.
Their stance has now softened, however, and Barcelona are preparing a £145m offer to finally bring Coutinho to the Nou Camp, according to the Daily Mail.
Asked to rate the chances of Coutinho remaining at Anfield beyond the current transfer window, Klopp said this week: ‘All the things I could say now would just create stories, so I have nothing to say about it.
‘It’s the transfer window, that’s all. There’s nothing I could say that would help me, the player or the club.’
The proposed £145m fee would be a record for a player without a release clause in his contract.
Liverpool, who are fourth in the Premier League, progressed to the fourth round of the FA Cup on Friday evening with a 2-1 victory over rivals Everton.
James Milner fired the Reds ahead from the spot but Gylfi Sigurdsson equalised midway through the second half as Everton looked destined to force a replay.
But £75m signing Virgil van Dijk completed a dream start to his career at Anfield with a late header to settle the affair.