The Premier League side are interested in adding the prolific Algerian striker to their new attacking ranks.
Sporting's Islam Slimani celebrates his goal against Braga
The Premier League side have been committed to strengthening their attacking roster this summer after a number of departures from the Liberty Stadium, and the 28-year-old is the latest target.
Bafetimbi Gomis, Andre Ayew, Alberto Paloschi and Eder have all left South Wales since the end of the previous Premier League season, leaving room for new additions up front.
Bafetimbi Gomis celebrates scoring for Swansea City with Andre Ayew
Spanish international Fernando Llorente has already signed for the Swans, while Borja Baston is expected to join him in making the move from La Liga, for a fee that will break the current transfer record.
But Slimani would break the reported £15 million deal for Baston (Sky), with Sporting having already rejected a figure higher than that.
The Algerian striker has had a phenomenal few seasons in Lisbon since arriving in 2013, with the last year full of particularly impressive performances.
Sporting's Islam Slimani celebrates his goal against Pazos Ferreira
Slimani smashed an astonishing 31 goals in 46 games over the course of the 2015-16 campaign, including 27 goals in 33 league matches. His three seasons in Portugal has seen him win the country's FA Cup in 2015 and the Portuguese version of the Community shield last season, and with every passing campaign the striker has become a more important figure and a more prolific goalscorer.
It has resulted in a number of links for his services, including several Premier League clubs, but it looks like Swansea and their manager Francesco Guidolin are the side most committed to recruiting him.
Swansea manager Francesco Guidolin