The 22-year-old winger has taken over Stewart Downing's mantle since the England international's summer departure for Aston Villa, and is Boro's leading scorer this season with nine goals to date.
However, his existing deal is due to run out at the end of the campaign and talks over an extension have yet to pay dividends but chief executive Keith Lamb said: "We have had no approach from Sunderland, nor would we welcome one. Adam is a Middlesbrough player and we want him to stay."
Sunderland-born Johnson, who will be able to talk to other clubs in January, has attracted the attention of a series of suitors - with the Black Cats and Everton among those to have run the rule over him.
But amid speculation that Steve Bruce is ready to firm up his interest with a £3million bid for the England Under-21 international when the transfer window opens in January, Boro are refusing to give up on one of their most successful Academy graduates.
Johnson has started all 16 of Boro's game to date this season, and passed up an opportunity to claim his 10th goal when he missed a penalty in Saturday's 1-0 home defeat by Plymouth.