The Sun says The Kop had an initial bid of £10 million knocked back by Ajax - but they returned with an improved £13.5 million offer last night.
And Ajax chiefs have admitted the left winger, 20, will be packing his bags for Merseyside.
Ajax's technical director Martin van Geel said: "Liverpool are very serious about signing Babel and have made an offer a lot higher than before. Because they want him so much, I think we will lose him."
Babel is still on holiday in Surinam after helping Holland's Under-21s win the European Championships. Yet he has made no secret of his desire to play in the Premier League and is convinced he can prove a smash hit.
Babel, who has also been interesting Arsenal, admitted: "Of course I'm ready but if the time is right or it's the right club is another matter.
"I'm still learning a lot at Ajax and they're a fantastic club. I'm still young and not in a hurry.
"We'll see what happens but for sure I want to play in the Premier League and again it's only a question of when."