Rafael Nadal, Roger Federer, Alexander Zverev and Dominic Thiem have already qualified for the prestigious event, to be held 12-19 November at The O2 in London.
But four other places remain, with Grigor Dimitrov, Pablo Carreno Busta, Sam Querrey and Kevin Anderson looking to boost their chances at the Erste Bank Open 500 in Vienna. Marin Cilic, David Goffinand Juan Martin del Potro will hope to do the same at the Swiss Indoors Basel.
Dimitrov might not have to wait until the new tournaments start on Monday; he could clinch his London debut on Sunday. The Bulgarian faces del Potro in the Intrum Stockholm Open final. With his second Stockholm title, the 26-year-old Dimitrov will book his place in London.
Delpo can't qualify on Sunday, but the Argentine has an outside chance of returning to the season-ending tournament (also 2008, 2009, 2012, 2013). If he repeats as Stockholm champion on Sunday, del Potro will be 470 points behind Carreno Busta, who currently holds the final qualification spot.
The Spaniard is looking to also book his first appearance at The O2. Carreno Busta is the fourth seed in Vienna, but has a tough draw. After facing a qualifier in the first round, Carreno Busta could face Fabio Fognini of Italy or Argentine Diego Schwartzman, who will try to win his second ATP World Tour title on Sunday at the European Open in Antwerp.
Cilic, the defending champion in Basel, needs to reach the semi-finals to officially start planning his third trip in the past four years to the Nitto ATP Finals. The second seed faces Stockholm semi-finalist Fernando Verdasco of Spain in the first round and could meet #NextGenATP countryman Borna Coric in round two. Eighth seed Mischa Zverev of Germany is the seeded favourite to meet Cilic in the quarter-finals.
Goffin, who was an alternate at the Nitto ATP Finals last year, is the third seed in Basel and might face Italian veteran Paolo Lorenzi in the second round and fifth seed Jack Sock in the quarter-finals. Anderson, the seventh seed, and Querrey, the eighth seed, will also try to make headway in Vienna.
Federer qualified for his 15th appearance at the Nitto ATP Finals months ago, and the Swiss will look to stay sharp in his hometown tournament, where he is a seven-time champion and making his first appearance since 2015.
The 36-year-old opens against #NextGenATP American Frances Tiafoe, who's looking to qualify for the inaugural Next Gen ATP Finals, to be held 7-11 November in Milan. Federer could then meet American Steve Johnson. A quarter-final with #NextGenATP Canadian Denis Shapovalov would be must-see tennis, but seventh seed Adrian Mannarino might prevent that. Goffin and Sock are the favourites to face Federer in the semi-finals.