Smith, however, remain optimistic his side can fend off the interest and recently told reporters for the London Evening Standard how now club has come close to offering his true market value.
“At the moment I do not envisage him leaving. No club has got anywhere near his valuation and I do not see that in the foreseeable future,” Smith said.
"I do not agree with what he did. I made it known to James how I felt about his decision and he understood my reaction. But we will not bite his head off. We will try and help him through the tough period."
Tarkowski was disciplined by the club after declaring himself unavailable for selection, incidentally missing Brentford’s recent 3-1 home defeat to renowned suitors Burnley.
The Baggies haven’t made any moves in the market this month, yet Tarkowski has seemingly impressed boss Tony Pulis despite the West Midlands side loaded with centre-back options.
The Owls are another said to be keen on the 23-year-old, though after signing Gary Hooper, Barry Bannan and Jack Hunt on permanent deals Wednesday’s transfer business could already be done.
Tarkowski has become a mainstay of the current Brentford side, and has established himself as one of the top defenders in the Championship this term.