Ana Ivanovic, Sam Stosur and Anastasia Pavlyuchenkova booked their spots in the semi-finals of the WTA Tournament of Champions in Sofia.
Heather Watson was forced to pull out of her quarter-final against British compatriot Johanna Konta in Barnstaple because of illness.
Novak Djokovic kept alive his slim hopes of regaining top spot in the world rankings by reaching the Paris Masters semi-finals.
Roger Federer beat Juan Martin Del Potro 6-3, 4-6, 6-3 to book his place in the last four of the Paris Masters.
Hosts Italy are huge favourites to beat weakened Russia in the Federation Cup final in Cagliari this weekend.
The line-up for next week's end-of-season ATP World Tour finals in London was completed on Thursday following Milos Raonic's exit from the Paris Masters.
Ana Ivanovic put herself in pole position to qualify for the semi-finals at the WTA Tournament of Champions in Sofia.
Andy Murray is not interested in going to the Australian Open unless he is in a position to win it.
Rafael Nadal moved into the third round of the Paris Masters with a straight sets victory over fellow Spaniard Marcel Granollers.
Roger Federer made sure of his qualification for the ATP World Tour finals with a straight-sets win over Kevin Anderson at the Paris Masters.
Simona Halep defeated Alize Cornet to claim a spot in the semi-finals of the Garanti Koza WTA Tournament of Champions in Sofia.
Rafael Nadal booked his place in the third round of the Paris Masters with a straight-sets win over fellow Spaniard Marcel Granollers.