According to Sky Sports, the Premier League club reportedly paid Sporting an additional one million euros on top of his 10 million euro ($11 million) exit clause for an early release.
Sporting issued a statement to the Lisbon stock exchange on Tuesday confirming United were willing to meet the release clause.
United pulled the plug on Erik ten Hag's two-year reign as manager on Monday after a 2-1 defeat by West Ham left them 14th in the Premier League.
Amorim quickly emerged as United's number one option.
United caretaker boss Ruud van Nistelrooy oversaw a 5-2 victory against Leicester in the fourth round of the League Cup on Wednesday.
"I came here as an assistant to help the club," said Van Nistelrooy. "I'm helping as long as I'm needed and in the future, in any capacity, I'm here to help the club build towards the future."
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Despite spending heavily in the transfer market in the summer, Ten Hag paid the price for United's poor start to the season.
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Prior to the win against Leicester, they had won just one of their past eight games in all competitions.
(Except for the headline, this story has not been edited by staff and is published from a syndicated feed.)