Rafael Nadal's dramatic 6-3, 6-4, 3-6, 4-6, 15-13 fourth round defeat against Luxembourg's 16th seed Muller was his fifth exit before the quarter-finals in his last five visits to the All England Club.
Sania Mirza and Barbora Strycova could next face the team of Martina Hingis and Chan Yung-jan in the Indian Wells women's doubles quarterfinals.
Rohan Bopanna is all set to clash with Novak Djokovic in the ATP Indian Wells Masters event while Leander Paes will be seen playing with Juan Martin Potro and the pair will be up against Gilles Muller and American Sam Querry.
Rohan Bopanna and Florin Mergea pair eased past Lukas Dlouhy and Jiri Vesely to enter the pre-quarterfinals of the Australian Open.
Janko Tipsarevic's inclusion in the 20th edition of the Chennai Open will see 10 players from world's top 50 list.
Andy Murray dropped the first set and was taken to a tie-break in the second before finally prevailing 3-6, 7-6 (7/2), 6-4 to reach the last four at the Wimbledon warm-up event for the fourth time.
Eighth seed Gilles Muller came from a set down to beat last year's finalist Eduard Roger-Vasselin 6-7, 6-1, 7-6.
The French - Alize Cornet and Jo-Wilfred Tsonga - will face the top-seeded Polish pairing of Agnieszka Radwanska and Grzegorz Panfil in Saturday's final.
Gasquet, who is ranked 10th in the world, won in a tight three-setter 4-6, 6-4, 6-4 in two hour 23 minutes to record his second victory over Kukushkin in as many meetings.
The 22-year-old fifth seed -- ranked 18 in the world -- took 1hr 41minutes to defeat her seventh-seeded opponent and 2011 US Open champion and record only her second win in five meetings with the Australian.
The Asian pair of Rohan Bopanna and Jie Zheng defeated unseeded Swedish-Hungarian duo Johan Brunstrom and Katalin Marosi 7-6(4), 3-6, 6-1 in one hour and 29 minutes.
Li led fellow seeds to opening victories on a calm day at normally blustery Devonshire park. Number three Angelique Kerber beat Sorona Cirstea 6-4, 6-4 while 2011 Wimbledon champion Petra Kvitova stopped Monica Niculescu 6-4, 6-1.
The second-seeded Chinese Li Na player will play 2011 champion Marion Bartoli of France in the second round.
Third seed Victoria Azarenka came from a set down to beat France's Alize Cornet 4-6, 6-3, 6-1 to reach the French Open fourth round on Saturday.
Top seed Samantha Stosur of Australia battled back from a set down to beat fourth seed Ana Ivanovic of Serbia on her way to the final of the joint ATP and WTA Kremlin Cup tournament here on Saturday.
Caroline Wozniacki advanced to the final of the Kremlin Cup on Saturday by beating Sofia Arvidsson of Sweden 6-3, 6-7 (4), 6-4.
Former No. 1 Caroline Wozniacki advanced to the Kremlin Cup semifinals by beating defending champion Dominika Cibulkova 6-2, 6-7 (1), 6-1 on Friday.
Indo-Spanish pair of Sania Mirza and Nuria Llagostera lost out to Americans Raquel Kops-Jones and Abigail Spears in the women's doubles quarter-finals of the Kremlin Cup here on Thursday.