Spurs boss Harry Redknapp is keen to add depth to his squad in the transfer window and sees Downing as a player who will give his side much needed balance down the left flank.
However, a Boro spokesman said reports of a third, £14.5m bid are way off the mark.
"That's not true, we have not had another bid," he said.
Downing, who has 21 England caps, signed a new five-year deal with Boro in February last year but has become increasingly frustrated with the club's inability to establish themselves in the top half of the Premier League.
Tottenham had an initial bid of £6m for the player rejected by Middlesbrough last week.
Manager Gareth Southgate has repeatedly insisted that local boy Downing would not be sold.
And Lamb shares the manager's view, saying on Monday: "We received a written transfer request from Stewart. It was considered before being rejected. "Prior to receiving the transfer request, we received an improved offer for Stewart from Tottenham.
"We immediately rejected this offer and informed Tottenham that Stewart Downing is not for sale and any further interest they have in the player will not be entertained."
Chairman Steve Gibson has also previously stated that no senior player would be leaving in the January transfer window.
Boro club captain Emanuel Pogatetz has called on Downing to remain focused on the club while he remains at the Riverside.
He's frustrated that the team is once again struggling in the Premier League
Stewart Downing Sr
"I don't know why he made this decision," he told BBC Tees.
"He has to stand up for this now and hopefully Stewart will still give his best for the club and perform well."
The Teesside club are under no financial pressure to offload one of their most prized assets but Southgate is not thought to have been previously aware of Downing's desire to leave.
But Downing's father, Stewart Sr, told Middlesbrough's Evening Gazette: "Stewart's been frustrated since the summer.
"When he signed his new contract he was assured that there would be significant investment in the squad.
"He feels that the club haven't matched those expectations. He's also frustrated that the team is once again struggling in the Premier League.
"Obviously as a Middlesbrough lad, he's thought long and hard about the decision to hand in a transfer request, and he also hopes the fans will understand his reasons for doing so."
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