The 21-year-old left-back, who thrived for the Swans last season, has become one of new manager Mauricio Pochettino's top transfer targets this summer.
A deal was agreed in principle for the Wales international to move to White Hart Lane last week, but Swansea chairman Huw Jenkins pulled out of the arrangement in search of a better offer.
Spurs, however, remain determined to capture Davies, who is valued by the Welsh club at £12million, and have begun fresh negotiations over a possible sale.
Swansea are thought to be very reluctant to sell a player who has made 71 appearances for the club since his debut in 2012.
Davies became a regular during the 2012-13 season following long-term injury to defender Neil Taylor, excelling under both previous manager Michael Laudrup and current boss Garry Monk.
Davies, who has 10 caps for Wales, signed a one-year contract extension last year that keeps him tied to the Liberty Stadium until 2017.
Tottenham, though, are in need of a left-back, with centre-back Jan Vertonghen filling in for the injured Danny Rose for large periods of last season.
Reports last month suggested they had moved for Sevilla youngster Alberto Moreno, but Tottenham are not interested in the Spaniard.