Wales captain Williams, 31, has joined the Toffees for an undisclosed fee, thought to be up to £12m with add-ons.
"It is definitely not something we wanted to happen," Jenkins told the Swans' website.
"But having turned down Everton's initial offer, Ashley expressed his desire for a new challenge."
Jenkins continued: "Taking into account the excellent service Ashley has given this football club over the last eight years, including as club captain, we reluctantly agreed to his request to join Everton after they made an improved offer.
"It goes without saying that we wish him the very best with his new challenge and thank him for his services."
Williams, whose contract is due to run until 2018, has played more than 350 games for the Swans since joining for £300,000 from Stockport in March 2008.
His transfer follows the departure of fellow centre-half John Stones from Everton to Manchester City for £47.5m on Tuesday.
Williams had not played in any pre-season matches having been given an extended break after captaining Wales to the semi-finals of Euro 2016.
Williams said: "I've had eight great years at Swansea and, for me, the new challenge was something I felt I needed.
"Swansea was very memorable for me and I would like to thank every single fan, my former and current team-mates and all of the Swansea family for their unconditional support and friendship.
"I wish everyone at Swansea all the best for the future and they will always hold a special place in my heart."
Chairman Huw Jenkins says Swansea were disappointed to see Ashley Williams join Everton but had "reluctantly" agreed to his transfer request.