Federer admitted it would be great if he could take on his compatriot Stanislas Wawrinka in their home country of Switzerland.
Wawrinka and Tiafoe played out a gruelling match which lasted over two and a half hours fetched the Swiss his fifth Basel quarter-final berth.
Tiafoe saved a Wawrinka match level with an ace of his personal within the ultimate recreation earlier than the house favorite bought the gang on its ft with a backhand victor down the road to seal the success. "But I played two very good matches this week so far and this is definitely the best I have ever played in Stadthalle".
Both players hugged at the net, with Tiafoe receiving a standing ovation from the 9,000-strong crowd. "I had to really raise my level, Frances put up a huge fight". "I was basically a point away from losing", said Federer.
"I managed to remain calm and not rush too much". "I hope Stan goes through, I always enjoy playing him", added Federer, who holds a 23-3 edge in the series against his close friend. But getting angry sometimes can actually help.
In other second-round action, the United States' Reilly Opelka edged sixth-seeded David Goffin of Belgium 6-7 (4), 7-6 (4), 7-5, and fourth-seeded Roberto Bautista Agut of Spain topped France's Richard Gasquet 6-2, 4-6, 6-3.
Goffin stands provisional tenth in the race to the eight-man season wrap-up with a final points-earning chance next week at the Paris Masters.
Rafael Nadal, Novak Djokovic, Federer, Daniil Medvedev, Dominic Thiem and Tsitsipas have already booked their locations within the ATP Finals, leaving simply two locations out there.