The Mirror says Leicester have made a £3million bid for Naughton.
While Spurs have not accepted the offer, they are willing to allow Naughton to leave this month - and Swansea are expected to table a counter offer.
The Foxes are having a busy January in the transfer market, having agreed a £9m deal for Croatian striker Andrej Kramaric - subject to a work permit - and have signed veteran keeper Mark Schwarzer, 42, from Chelsea.
Boss Nigel Pearson has also identified Naughton as a player who can help Leicester avoid relegation, but the midlands club may have to up their bid.
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