A delegation from Spurs saw the 23-year-old score in a 1-1 draw against Le Havre two weeks ago and plan to watch him again before January.
Briand has two years left on his contract and his club value him at £8m, having already turning down a £6m bid from Paris Saint Germain in the summer.
The asking price could be reduced to £7m as Rennes are thought to have pushed the figure up because they were so reluctant to sell to another French club.
Briand was watched closely by Portsmouth towards the end of last season and has recently been called up to the full France squad by Raymond Domenech.
He can also operate on the left wing and his style has been compared to Thierry Henry.
He is believed to be the fastest player in the French league after having his 100 metre sprint time clocked at just over 10 seconds.
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