The 22-year-old winger has taken over Stewart Downings mantle since the England internationals summer departure for Aston Villa, and is Boros leading scorer this season with nine goals.
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.
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 £3m 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.
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. Johnson has started all 16 of Boros games, and passed up an opportunity to claim his tenth goal when he missed a penalty in Saturdays 1-0 defeat to Plymouth.