The Mexico international has a year left to run on his contract at White Hart Lane but has grown frustrated at his lack of first-team opportunities, having had loan spells with Ipswich, Galatasaray and Racing Santander. The 23-year-old is set to hold talks over his future with Spurs chairman Daniel Levy following Mexico's ongoing United States tour.
Rodgers was in attendance as the forward turned in an excellent display for Mexico in their 2-0 win over Wales in New York on Sunday, with his teasing runs causing the Dragons defence all manner of problems.
It is understood that any move for Dos Santos would be a loan deal, with Swansea set to break their transfer record to bring midfielder Gylfi Sigurdsson to the club on a permanent basis.
The Swans have agreed a fee of £6.8million with Hoffenheim for the Iceland international, who scored seven goals in 18 appearances while on loan to the Swans during the second half of their maiden Barclays Premier League campaign.
Sigurdsson, who was keen to stay at the Liberty Stadium, has already agreed personal terms and will undergo a medical later this week before he completes his move.
It will mark a full-time return to English football as the 22-year-old previously had two years at Reading between 2008 and 2010 prior to his £6million-plus move to the Bundesliga.
Sigurdsson was named Hoffenheim's player of the year in his first campaign, despite starting only 13 matches, but having missed much of pre-season he fell out of favour and was loaned to Swansea in January, making an instant impression. The deal will almost double the club's previous highest outlay of £3.5million for striker Danny Graham last summer.
"Swansea City can today confirm a transfer fee and personal terms have been agreed for Hoffenheim midfielder Gylfi Sigurdsson," said a statement on the club's official website.
"A club record offer of £6.8million has been accepted by the German club, and the 22-year-old is expected in Swansea later this week to undergo a medical before the move is confirmed. The club are delighted to have agreed terms with both player and club following Sigurdsson's hugely successful loan spell last season."