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 Middlesbrough 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.”
A goal apiece from James Tomkins and Luka Milivojevic handed Crystal Palace a crucial 2-0 win over Huddersfield at the John Smith's Stadium on Saturday evening.
Manchester United bounced back from their Champions League exit as they beat Brighton 2-0 at Old Trafford on Saturday night to reach the FA Cup semi-finals.
Tottenham reached the FA Cup semi-finals after they cruised to a 3-0 victory over Swansea in their quarter-final clash at the Liberty Stadium on Saturday.
Premier League basement club West Brom squandered the lead as they suffered a 2-1 defeat to Bournemouth at the Vitality Stadium on Saturday.
Chesterfield boss Jack Lester felt his side were unlucky not to claim all three points after their 1-1 draw at Cheltenham.
New Swindon manager Phil Brown revelled in his side's victory at Cambridge in his first game in charge.