Amorim, 39, is one of the most sought-after young coaches in Europe and had reportedly been in the running to replace Jurgen Klopp at Liverpool over the summer.
Man Utd would need to pay a release clause for the Portuguese coach, but are willing to do so, according to Sky Sports News.
Amorim led Sporting to their first league title in 19 years in 2021, and won his second with the club last season.
So far this season his side is unbeaten and tops Liga Portugal with nine wins from a possible nine.
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Manchester United fans said the club needs to move on without Erik ten Hag and it was ‘about time’ to sack him following a disappointing start to the season.
Man Utd appear to be moving quickly to try and appoint a successor to Ten Hag as the club languishes in 14th after a poor start to the season.
Former Red Devils striker turned assistant coach Ruud van Nistelrooy has replaced Ten Hag on an interim basis.