The 27-year-old, who completed a £5million switch from Swansea to the Stadium of Light on Thursday afternoon, was booed by sections of the home support conscious of his Newcastle allegiance as he came on for Leon Britton.
That prompted Boro to see if there was any possibility of taking their former trainee back to Teesside.
Manager Mowbray told the club's official website: "I would have to say to our supporters that we have never been to the point of making an offer of £5million.
"I think our fans know we don't have that sort of resource and that offer was never made.
"I did hear on the radio on Tuesday night that when Danny got on the pitch at the Stadium of Light, it wasn't a favourable reception for him, which just in my mind threw up the potential of an opportunity, really.
"If he was committed to coming back to the north east and if Sunderland wasn't an option, then maybe something could have been done."