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He will be replaced in the draw by Zizou Bergs.
Murray held his nerve in the deciding 10-point tie-break, closing out the match with incredible fortune as his bunted forehand return caught the net and left Berrettini no chance.
United States outplayed Italy 3-0 in the final.
Italian Lorenzo Musetti squandered five championship points in the second set but came back to stun top seed Carlos Alcaraz and win his first ATP Tour title in Hamburg.
Sweden's Elias Ymer replaced Berrettini in the men's draw.
A win for Nadal in Sunday's final will break a three-way tie with Novak Djokovic and Roger Federer and give him sole owenership of the record for most Grand Slam singles titles.
The 25-year-old, who will make his second appearance at the ATP Finals, defeated Australia's Alexei Popyrin 7-6(3), 6-3 in the opening round of the Vienna Open.
Novak Djokovic beat Italian Matteo Berrettini 6-7(4), 6-4, 6-4, 6-3 to win his sixth title at Wimbledon.
Nadal put pressure on the underdog Italian's serve from the start but Berrettini saved six break points to set up a first set tie breaker.
The six-time champion was not at his fluent best on Tuesday but ultimately had too much for Berrettini, who won just three games against Novak Djokovic in his opener.
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