Hammers boss Slaven 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 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 manager [Brendan Rodgers] has been a great boss, he’s been at Liverpool, and Moussa will learn a lot from him and the coaching staff.”
Dundee United manager Csaba Laszlo refuses to give up on promotion despite a nightmare run in the Scottish Championship.