The Baggies sit bottom of the table, 10 points from safety and with just three league wins all season to their name.
Evans has been one of the club's standout performers since his 2015 move from Manchester United and has been linked with the likes of Manchester City and Arsenal in recent months.
Spurs, meanwhile, are considering whether to cash in on Toby Alderweireld, who could reportedly leave North London for £25m in 12 months' time because of a clause in his contract.
According to The Telegraph, Mauricio Pochettino is monitoring Evans, an experienced player who would be available for £3m due to a release clause should Albion drop into the Championship.
The Northern Ireland international has made 93 appearances for West Brom and has also been capped 67 times for his nation.