Spurs are said to be willing to sell the American international, despite him only having made the short trip to White Hart Lane from Fulham last year.
This has alerted Baggies chief Steve Clarke, who will have to make do without Romelu Lukaku next season, after the burly Belgian returned to Chelsea following a successful loan spell at The Hawthorns.
He is hopeful that Dempsey can add some real Premier League know-how and vital goals to his attacking set-up in a bid to improve on an impressive 2012/13 season.
However, Tottenham are keen to recoup at least the £6m they paid for the 30-year-old, which, allied to his wage demands, could price West Brom out of a move.
Although he failed to take many headlines in north London last term, Dempsey netted an impressive 12 times, making him one of the club’s major sources of goals.
When this is tallied alongside his 23 strikes in the previous campaign with Fulham, the Texas-born attacker can claim to be one of the best goal-getters in the English top flight.
West Brom have already been active in the transfer marker, with the experienced Nicolas Anelka having already arrived.
West Brom fans, should your side spend big for Clint Dempsey?
Tell us what you think below!
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