Federer, 39, was pushed to the limit in his first match back on Wednesday when he beat Briton Dan Evans in three sets but the Swiss looked well in control in the opening set against the unseeded Basilashvili.
World No. 1 Jannik Sinner stayed cool to reach the second round of Wimbledon while several of his fellow seeds including Italian compatriot Lorenzo Musetti crashed and burned on a sizzling Tuesday.
Home favourite Jo-Wilfried Tsonga sauntered into the semi-finals of the Lyon Open with a 6-0, 6-4 victory over Russian youngster Karen Khachanov on Thursday.
Nikoloz Basilashvili has denied allegations that he attacked Neka Dorokashvili in front of their five-year-old son and he has been released on 100,000 lari ($26,000) bail.
Mischa Zverev took some of the limelight back from younger brother Alexander when he beat second seed Kei Nishikori to reach the Geneva Open final on Friday.
Indian qualifier Prajnesh Gunneswaran achieved the biggest win of his career when he stunningly knocked out World No 18 Nikoloz Basilashvili at the Indian Wells ATP Masters event, continuing his dream run.
Rafael Nadal overcame a temporary blip on his way to victory at the ATP Cup on Saturday
In his pre-tournament news conference in Toronto on Sunday, Nadal said a recurring foot issue he has been dealing with since 2005 returned at the French Open, where he fell to eventual champion Djokovic in the semi-finals.
Clearly Nikoloz Basilashvili had come to face Rafael Nadal with a plan but, like Mike Tyson said, everybody has a plan until they're punched in the face.
Australian Nick Kyrgios was expelled from the tournament after going wild in the third set of his match against Casper Ruud.
World number one Novak Djokovic produced a majestic display in a 7-5, 6-3 victory over American John Isner on Thursday to extend his winning streak in Asia.
Britain's Kyle Edmund reached the quarter-finals of a Grand Slam for the first time after coming from a set down to oust Italian Andreas Seppi 6-7(4), 7-5, 6-2, 6-3 in the fourth round of the Australian Open on Sunday.
Pummelling the ball with such force that it appeared distorted into an oval shape at times, the nine-times French Open champion was able to toy with his opponent until ready to finish him off.
Former world number one Novak Djokovic showed signs of a return to form with an impressive 7-5, 6-4 victory over Japan's Kei Nishikori in the opening round of the Madrid Open on Monday.
Serena Williams returns to the Arthur Ashe Stadium on Sunday
Serena faces hard-hitting Giorgi in first round while former World No Hewitt play his final grand slam
Top seed and defending champion Roger Federer battled past in-form Russian Daniil Medvedev 6-4, 4-6, 6-4 to reach the last-16 at the Shanghai Masters on Wednesday.
Garbine Muguruza was relieved to finally avoid tripping at the last hurdle and end her trophy drought
Images from matches played on Day 5 at the Australian Open in Melbourne Park on Friday
Eight-time Wimbledon champion Roger Federer, who turns 40 this year, takes on Frenchman Adrian Mannarino in round one.
Photos from Day 1 of the Australian Open matches played at Melbourne Park on Monday
Paula Badosa survived a stiff challenge from Victoria Azarenka in the Indian Wells final, while Cameron Norrie battled back to defeat Nikoloz Basilashvili.
Djokovic first played Monfils at the US Open in 2005 but Tuesday's match was only their second encounter on clay and he needed only one break point to clinch the first set.
Images from Day 1 of the Wimbledon tennis championships in London, on Monday.
Roger Federer was made to work hard by Fernando Verdasco before beating the Spanish veteran 6-3, 3-6, 6-3 on Wednesday to reach the quarter-finals of the Dubai Duty Free Tennis Championships.
Novak Djokovic extended his unbeaten start to the season to 15 matches as he swatted aside German veteran Philipp Kohlschreiber 6-3 6-1 with ease on Wednesday to reach the Dubai Tennis Championships quarter-finals.
IMAGES from Day 7 of the US Open on Sunday
Home favourite Kei Nishikori defeated Frenchman Richard Gasquet 7-6(2), 6-1 in the semi-finals of the Japan Open on Saturday, setting up a final against Russian qualifier Daniil Medvedev.
Wimbledon champion Garbine Muguruza suffered a 6-2, 6-4 defeat at the hands of unseeded Czech Barbora Strycova in the second round of the Nature Valley Classic in Birmingham on Thursday.
Summary of sports events and persons who made news on Thursday
A tenacious Roberto Bautista Agut battled back to beat world number one Novak Djokovic 1-6, 7-5, 6-3 and advance to the quarter-finals of the Miami Open on Tuesday as the Spaniard claimed his second win over the Serb this year.
Images from Day 5 of the Wimbledon tennis championships played in London, on Friday.
Karolina Pliskova lost her cool over a line call and bashed a hole into the umpire's chair with her racket after suffering a 3-6, 6-3, 7-5 defeat by Maria Sakkari in the Italian Open second round on Wednesday.
Images from Day 5 of the 2021 French Open, in Paris, on Thursday.
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