Downing has been a long-term target for Tottenham, and new Spurs boss Harry Redknapp wants to bring the international player to north London before the end of January's transfer window.
However, Lamb was left fuming after Tottenham's initial bid for the 25-year-old came way less than what Boro expected.
"It is a derisory offer," Lamb told the News of the World.
'Who on earth do they think they are? They bleated away when Dimitar Berbatov left for Manchester United in the summer - but they sold at top dollar.
Bulgarian striker Berbatov was sold by Tottenham to United in the summer for a massive transfer fee of £30.75 million.
"Then they want to snatch other players for derisory offers and try to unsettle players. That is what has happened here. Their offer is a joke.
"Why would we sell our best players to our nearest rivals? Stewart is going nowhere."