The Team GB midfielder joined the Reds from Swansea City last week after securing a £15m move. The Wales international, who will be reunited with former Swans boss Brendan Rodgers, admits that they have enjoyed a turbulent few years together.
"The last couple of seasons have definitely been a whirlwind," Allen told Sky Sports News.
"First and foremost, we got promotion to the Premier League which was something I hadn't imagined so quickly and then off the back of last season to find myself where I am today has been an incredible journey.
"I'm very grateful for the times I've had and I'm looking forward to more magic moments in the future."
The 22-year-old was a key part of Swansea's success last season.