The Netherlands international has spent the last two seasons on loan at the Liberty Stadium from La Liga club Villarreal, enjoying League Cup success and a UEFA Europa League campaign.
Jenkins is eager to sign the former Feyenoord man for a third time after his latest spell with the Premier League club came to an end.
"We will talk to his club and agent to see what his thoughts are - as we did last summer," Jenkins told the club's official website.
Swansea appointed Garry Monk as manager on a permanent basis last week following a spell as interim boss after Michael Laudrup's departure in February.
And Monk has already set about shaping his squad for his first full season at the helm, with strike duo Leroy Lita and David Ngog leaving the club.
Lita arrived in South Wales from Middlesbrough in August 2011 but failed to establish himself as a regular and was loaned out for spells at Birmingham City, Sheffield Wednesday and Brighton and Hove Albion.
Former Liverpool man Ngog was signed from second-tier Bolton Wanderers on a short-term contract in January and joins Roland Lamah and Alvaro Vazquez in leaving the club.
The pair, loaned from Osasuna and Getafe respectively, saw their deals end at the end of this season.
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