The Welsh international signed a long-term contract at Anfield on Friday, in a deal Goal.com understands to value at £45,000-a-week salary with a £2 million signing-on fee.
Allen had been part of the setup at the Liberty Stadium since the age of nine, and Jenkins lamented the midfielder’s decision to take up a new challenge elsewhere.
“It is with great reluctance that we have agreed to the transfer of Joe to Liverpool,” Jenkins told the club’s official website.
“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.
“We are very lucky at Swansea City that football decisions are taken without any financial pressure to sell. But that also makes some decisions even more disappointed when they happen like this.
“However, as always, we will deal with those disappointments in the right manner and we wish Joe all the very best from everyone at Swansea City. It’s been a pleasure watching him grow and develop as a footballer and a person, while his professionalism and commitment to the cause has never been in doubt from the very first day he joined us. the door will always be open for him here.
“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.’’
Allen, himself, thanked Swansea fans for the support during his 13-year stay at the club upon his departure, insisting that the lure of Liverpool was too much to resist.
“Having come through the ranks and been part of the Swansea City for so long I have built up a massive attachment to the club and the fans,’’ explained Allen.
“I’ve really enjoyed my time at the club and will always be eternally grateful to the fans for the support they’ve shown me over the years.
“I’ve always been ambitious and while it is hard to leave a club I love, this is a fantastic challenge for me that I had to take. Having been with the club for so long, it’s a fresh and new challenge; a new chapter I needed to get started.
“I wouldn’t have changed my time at the club for the world and I’m proud to have been part of something that has been so exciting over the last few years. The club has been moving in the right direction and I’m sure it will continue to do so without me.
“I’m going to miss a lot of friends and family in Swansea, but I would like to wish the staff, players and supporters the very best for the future.’’
The Wales international made his league debut for the Swans in the final game of the 2006-07 season as a substitute against Blackpool, before going on to make 129 league appearance.
His impressive displays earned the Welsh speaker a call up to the recent GB Olympic team.