Jurgen Klopp's side limped out of the competition at the Bernabeu in a 6-2 aggregate defeat.
The Reds, beaten by Real in the 2022 Champions League final, saw their last chance of silverware this season disappear amid their struggles this season.
They also face a fight to even get back into the Champions League and are currently six points behind fourth place with 12 games left.
Their midfield has been a huge problem and Borussia Dortmund and England star Bellingham has been identified as a key target.
Any deal would likely smash a club, and potentially British, record transfer - while the player is set to decide his future after the upcoming international break.
They will also face a battle with the likes of Real Madrid and Man City in the summer.
However, former Manchester United defender Ferdinand has warned that Bellingham may be influenced by what he saw in the Champions League clash between the two sides.
Ferdinand told BT Sport: "I just think if he's watching that tonight, he's sitting there thinking if Real Madrid and Liverpool are in for me, there's only one place I can go, and [Liverpool] don't finish [in the] Champions League [places] as well maybe.
"As a player, you look at it cold and go 'I see more trophies here'."
Klopp admitted his side had not showed enough to deserve to go through in the Champions League over the two legs.
"It's true. We came here with the backpack we had with the three goals difference, and you need a special performance and we did not show a special performance tonight," he said.
"In moments it was a good performance but Real Madrid were the team in control for most of the game. They had the better chances.
"Ali [goalkeeper Alisson Becker] had to make two sensational saves to keep us in the game and that's the reason [we lost].
"If you want to get through, you need to be outstanding. If you want to win the game, you need to be really good, and Madrid was the better team and that's why the right team went through to the next round."