The 22-year-old, who played alongside the Welshman in Stuart Pearce's Team GB side at the London Olympic Games, completed his move to Anfield for a reported £15m last week.
"I spent the last three weeks hounding Joe Allen to go to Liverpool," Liverpool's official website quotes Bellamy as saying. "They have got an immense player.
"Joe, Brendan [Rodgers] and Liverpool will all be great winners out of this. It has been a pleasure to spend the last few weeks with him."
Meanwhile, Bellamy finalised a return to Championship side Cardiff City last week after spending a season at Liverpool.