Dawson's status as regular starter at Spurs was placed in doubt after he was left out of the squad for Saturday's opening day Premier League defeat at Newcastle.
Rangers are in the market for another centre-back, with their need seemingly becoming more pressing after the weekend's 5-0 home thrashing by Swansea.
Dawson had long been earmarked as a successor to Ledley King as Tottenham captain and he skippered the side during pre-season.
But the 28-year-old, who moved to White Hart Lane seven-and-a-half years ago, made just 13 appearances last term thanks to injury and it has been suggested he may now not be a key part of new manager Andre Villas-Boas's plans.
It is also understood QPR defender Matthew Connolly is on the brink of sealing a move to Cardiff, with a fee having been agreed between the two clubs.