The Blues had a £5million offer for the 22-year-old rejected last week but are now thought to have contacted the Teesside outfit once more with a substantially improved bid.
Boro boss Gordon Strachan has stated he expects Johnson to still to be at the Riverside after Monday's 5pm transfer deadline, but chairman Steve Gibson might be reluctant to turn down increased terms for a player who will be out of contract at the end of the season.
A product of Boro's highly-rated academy, despite being born in Sunderland, Johnson remained with the club following their relegation from the Premier League last term and it was anticipated he would be part of the club's push for promotion. Instead, as results have proved difficult for Strachan to come by, Johnson has found himself linked with several top-flight clubs.
However, he is said to have informed Boro officials last week of his desire to hear what City are offering and that is only likely to intensify as Monday's deadline draws near.