While Novak Djokovic - who Nadal ousted at the top of the rankings last week - started in dominant fashion against Matteo Berrettini, Nadal suffered a similar fate to that of Roger Federer, who went down to Dominic Thiem in his opening match.
The other two players, Roger Federer and Dominic Thiem in the same group are going to lock horns later on Sunday and Djokovic is slated to meet Federer for the 49th time this week.
The Swiss 20-time Grand Slam victor recovered quickly from being broken in the first game of the match but the Austrian fifth seed repeated the dose in the 11th game and took the opening set 7-5.
The 38-year-old Swiss, bidding for his seventh title at the event but his first since 2011, struggled to handle to Thiem's aggressive game as he went down 7-5, 7-5.
The Austrian stuttered while serving for the match, forced to save two break points, but he held his nerve to take the set 7-5 and seal victory on his second match point.
"I've got to recover, I've got make sure I play better than today, and hopefully I can win that match (against Berrettini)".
"I thought he played well. mean, the start definitely didn't help".
"My forecast, it's a normal tournament from here on forward", 20-time Grand Slam champion Federer, beaten by Djokovic in an epic Wimbledon final in July, said. I'm not allowed to lose anymore.
Nadal, takes on defending champion Alexander Zverev in the later match in Group Andre Agassi with major questions over his fitness since he withdrew injured from the Paris Masters at the semi-final stage. "I don't see things like this".
The Greek 21-year-old beat the Russian 7-6 (7-5) 6-4 at the O2 Arena.
Djokovic needs to at least reach the final to stand a chance of overhauling Nadal.
If the Serbian does claim the year-end number one ranking for a sixth time, he would match the record of Pete Sampras and victory at the end-of-season event would equal Roger Federer's mark.
The second-seed Serb, who has been drawn in the Bjorn Borg Group, needed just 64 minutes to beat his less experienced opponent.
"It's a victory that I craved for a long time now, and it's great that it came at this moment", Tsitsipas said.
"That's a strong statement, I never said I don't like him", he said before describing an incident in a match against Medvedev in Miami previous year when words were exchanged.
Both players are making their debut at the year-end tournament for the world's top eight players, which features a round robin before the semifinals.