The stocky winger shone for Switzerland over the summer at the World Cup, helping to fire his nation into the knockout phase – where they were beaten by eventual finalists Argentina.
Liverpool were subsequently linked with him, but despite claims that Shaqiri wanted to leave Bayern, he stayed put to battle for his first-team future.
Reports have since suggested that a lack of game time has forced the 23-year-old to alter his stance, which has drawn United into the running.
The Bundesliga champions are understood to be ready to sell if the player really wants to leave in January, leading to interest Wolfsburg.
The German side – backed by automotive giants Volkswagen – are keen to offer the wide-man a fresh chance, and staying in his current surroundings appeals to the former Basel ace.
Wolfsburg are also flying high in their division, tailing only Bayern Munich, and look odds on to qualify for next season’s Champions League.