The English newspaper reports that Blues boss Alex McLeish, who coached Fletcher with the Scotland national team, has attempted to lure his compatriot to St. Andrew's.
The Midlands club were relegated last season but are intent on bouncing straight back up to the Premier League, and hence have singled out some fairly lofty transfer targets.
Fletcher has already been sounded out by a number of top-tier teams, including Everton and Sunderland, as he is stuck behind the likes of Paul Scholes, Michael Carrick, Owen Hargreaves and Anderson in United's midfield.
However, the 24-year-old has made plain his desire to remain at Old Trafford, meaning McLeish and co will have to look elsewhere to bolster their promotion bid.