The right and left sided defenders have been two of the Premier League’s standout performers this term, providing a number impressive displays.
David Moyes is known to be in the market for full-backs this summer, with Patrice Evra set to retire and a top class option on the opposite flank to the Frenchman having been missing for some time.
The Scot is said to have around £100m to spend, and is ready to dedicate £45m of that to snapping up Shaw and Coleman.
Shaw is believed to be his preferred target, with the need for a left-back urgent.
The 18-year-old is also wanted by Chelsea, but United are confident that they can lure him to the North West.
Coleman has also emerged as a candidate with the Irishman having been one of the surprise performers of the season so far.
The 25-year-old is rated at around £20m, and Moyes is hopeful that his former club will open to doing business with him.