The promoted Swans are looking to strengthen between the sticks ahead of their debut Premier League campaign following Dorus de Vries' exit to Wolves.
The Welsh side have already acquired the services of Benfica keeper Jose Moreira, but Jenkins is keen to further bolster their resources.
Stockdale, who largely took a backseat to number one Mark Schwarzer at Craven Cottage last season, has said he would be open to a Liberty Stadium move.
But after enquiring with Fulham over the 25-year-old, Jenkins says it appears to be a dead end, telling Sky Sports News: "We've made enquiries there. It doesn't seem there's anything on the Fulham side that they want to make a deal happen, so it's looking unlikely that that's going to happen.
"But we're pursuing other options and we hope we're going to get something concluded again over the next seven to 10 days."
Fulham want a 24-hour recall clause on any deal involving Stockdale, but that is not permissible in Premier League loans - so he is set to join a Championship club instead.
Swansea have already completed three new signings this summer, but Jenkins says they still have business to do in the transfer market.
The chairman, who has confirmed Newcastle winger Wayne Routledge as a target, added: "We're definitely looking to bring in a few more.
"Whether that will be three, four players, we'll have to see.
"If our targets become available and we can get them out of the clubs they're at, then obviously we're going to try and do that so we can be competitive when the season starts."
Chelsea won the Europa League on Wednesday. The Blues defeated Benfica 2-1 on injury time goal from Ivanovic.
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