Hammers boss Bilic is desperate to add to his front line and it was reported last week that they had made a £20m offer for Dembele who only arrived at Parkhead back in the summer from Fulham.
Hoops skipper Scott Brown has urged the frontman, who has bagged 19 goals in 36 games, to stay where he is as he could be involved in a relegation scrap in the Premier League.
“Moussa has been on fire in the last five or six months,” Brown told reporters.
“Fair play to him, he does it in the big games as well and it’s been great for us to have him and hopefully we’ll have him for another good six months or so, anyway
“He’s going to learn a lot more under the gaffer than probably, without being disrespectful, going to someone who is fighting relegation.“
The Hammers are seven points clear of the drop zone but are desperate to strengthen and have also failed in bids for Sunderland striker Jermain Defoe and Hull winger Robert Snodgrass, although they hope to sign £15m-rated Scott Hogan this week.