Southampton coach Mauricio Pellegrino "would have happily sold" defender Virgil van Dijk to Liverpool, the Times have reported.
The Reds failed in their attempt to sign the Dutch international, with Southampton refusing to negotiate at any price.
The Saints had complained of an illegal approach from Liverpool, after which they apologised and issued a statement "ending all interest" in the player.
Van Dijk publicly expressed his desire to leave, but no move materialised and he remained at Southampton.
And it has now emerged that manager Pellegrino would have been willing to see the centre-back leave, having grown frustrated with his attempts to force through an exit.
It remains to be seen whether Liverpool will rekindle their interest in Van Dijk in January, although Pellegrino described the defender as "one of the best players in the Premier League" after his display in Saturday's 1-0 win over Crystal Palace.
"Around the player there is a lot of interest," he told Sky Sports. "You know how football is, it's a big business. We have to accept that. Around him it is the same situation. Now we have to leave aside this situation. Now everything is normal," Pellegrino said.
"Now, for me I've got this group of players and I will try to push them, to work with them, to help them, to improve every single day.
"We never know how the market will be in January. We can bring in some players, every single squad can improve. We have got people who are working to improve our squad all the time.
"And for the other teams it's the same. They will push to bring the best players and Virgil is one case. He's one of the best players in the Premier League.
"It's normal that the biggest teams in the Premier League want to buy him."