Manager Erik ten Hag is stepping up his desire to strengthen his attacking options and is also linked with an interest in PSV star Cody Gakpo, having conceded defeat in his pursuit of Ajax ace Antony.
Ten Hag was keen to bring the Brazilian with him from Amsterdam to Old Trafford this summer, having played a key part in his Ajax team that won back-to-back Eredivisie titles, but the Dutch giants' demand on £80million has forced the manager to look elsewhere.
Liverpool have also been linked with an interest in Sane with Jurgen Klopp a fan, but the Daily Mirror claim it is United who have made a surprise enquiry for the German star.
Sane has struggled for regular game time at Bayern Munich, making 22 Bundesliga starts last season, and the arrival of Sadio Mane from Anfield this summer has pushed the 26-year-old further down the pecking order.
It is claimed new Man United boss Ten Hag is happy with his options in attack with Anthony Martial, Jadon Sancho and Marcus Rashford clicking in pre-season.
But the continued doubts over Cristiano Ronaldo's future and the injury to Martial that will rule him out of United's Premier League opener against Brighton on Sunday has highlighted the lack of options across the front line.
Sane is only one of United's targets, with Gakpo also on Ten Hag's radar as the summer transfer window approaches its final few weeks.
The 23-year-old came through the ranks at Eindhoven, was coached by United legend Ruud van Nistelrooy in the club's academy and is now managed by the former striker in the PSV first-team.
Gakpo was honoured as the Dutch footballer of the year last season, following a bright season which saw him score 21 goals in all competitions, despite missing 12 games at the start of the campaign through injury.
With Ajax demanding £80m for Antony, it is claimed Gakpo would be half the price and available for around £40m - a decent price for one of the most talented young players in European football.
Starting his career as a wide man, Gakpo is now a central striker but would be a versatile signing for Ten Hag, with the PSV player and Man United player also represented by the same agency firm.