Boro are resolved not to sell for anything less than £15m, despite manager Gareth Southgate being told he must sell before he can buy.
Southgate does not want to lose one of his star players, especially for such a "derisory offer", and Boro chief executive Keith Lamb said: "Who on earth do Spurs think they are?
"They bleated away when Dimitar Berbatov left for Manchester United in the summer but they sold at top dollar.
"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."
Fellow midfielder Gary O'Neil is also being courted by other Premier League clubs, with Portsmouth the most interested.
Lamb said: "Portsmouth are at it as well, sniffing around Gary O'Neil, trying to get him through the back door.
"We're close to Spurs and Portsmouth in the table. Why would we sell these clubs our best players?"