Allen completed a move to Anfield on Friday, ending an association with the Swans that began at the age of nine.
Jenkins told the club's official website: "It is with great reluctance that we have agreed to the transfer of Joe to Liverpool. As chairman it is always a bitter disappointment when a young player who has been with us for a length of time decides his future lies elsewhere.
"Now, as we always do when a manager or player leaves us, we will re-focus and galvanise ourselves as a football club and make sure we continue to strengthen the squad ready for the start of the new Premier League season."
Swansea have now lost Welsh international Allen and manager Brendan Rodgers to Liverpool during a difficult summer at the Liberty Stadium.