Nottingham Forest also failed in their attempts to sign Swans star Pratley.
"There was serious interest in Leon, Darren and Angel," revealed Swans chairman Huw Jenkins.
"But we resisted the temptation to leave them go, taking into account the high value we place on the players as individuals in the squad."
Wigan boss Roberto Martinez is an admirer of Britton and wanted the 27-year-old playmaker to follow him to the Premier League and the DW Stadium.
Paulo feels it is in the club's best interests to keep everyone together until end of the season at least Swansea chairman Huw Jenkins Britton, who is out of contract at the end of the season, was subject of a deadline day enquiry that the Championship play-off hopefuls rejected.
Spanish defender Rangel, who is also out of contract in the summer, has long attracted top-flight interest.
But Fulham boss Roy Hodgson made a firm enquiry late on deadline day after the Cottagers signed another full-back, Nicky Shorey from Aston Villa, on loan until the end of the season.
Swansea's Championship play-off rivals Forest have been heavily linked with a £1.5m move for key midfielder Pratley during January but their enquiry was rebuffed by the fifth-placed Swans.
Jenkins is "delighted" the window passed without the Welsh club losing any of their key players while Swans boss Paulo Sousa allowed out-of-favour striker Stephen Dobbie to join Blackpool on loan until the end of the season.
"Paulo feels it is in the club's best interests to keep everyone together until end of the season at least. to help us achieve a position as high as we can," said Jenkins.
"All of the players understand our position to retain their services this season. They know how important they are to us and are fully committed and loyal to Swansea City."
Former Arsenal youngster Britton is one of Swansea's longest-serving players after joining from West Ham United in 2003 - after a brief loan spell - as a teenager.
Give your views on Swansea City The future of fans' favourite Britton, who roomed with Wigan boss Martinez when the pair were together at Swansea, was under intense speculation for over a month.
It was sparked after he conceded he was "surprised and disappointed" that Sousa made public details of his contract negotiations.
The Swansea boss revealed Britton, who has made more than 300 appearances for the Welsh club, would become one of the best paid players in the club's history if he signed a new deal.
Britton has also previously admitted he has been unsettled by the turmoil of his recent split with his daughter's mother who has moved back to his native London.
Martinez has already signed one-time Swans stars Jason Scotland and Jordi Gomez for Wigan and the Spaniard, who has previously stated Britton has Premier League quality, is interested in signing Britton.