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.