The Welshman, who was part of the side that lost 3-0 to West Bromwich Albion at the Hawthorns, is hoping to play in Thursday's Europa League tie against Hearts and Sunday's visit of champions Manchester City in the Premier League.
"It was brilliant to make my debut. It wasn't the ideal debut in terms of the result, but it was one of those days that didn't go for us and we have to move on from that," the 22-year-old told the club's official website.
"It could be a very special week for me, getting the chance to play in Europe for Liverpool and then hopefully my Anfield debut against Manchester City, and we hope at the end of both of those games it will be a nicer feeling. We will certainly give everything we have got for our supporters."
Allen joined the Reds from Swansea City for £15m earlier this month.