The 22-year-old was in the directors' box at Anfield to watch the Reds' Europa League 3-0 defeat of FC Gomel, which wrapped up a 4-0 aggregate success.
And speaking after the match, Rodgers revealed he was expecting to complete the signing before the weekend.
"I hope to conclude the deal in the next 24 hours. It is imminent," the former Swansea boss said after the victory over Gomel.
Sources emanating from the United Kingdom believe Allen will has already agreed personal terms with the Reds, and will undergo a medical at the club on Friday morning.