There is no doubt that Churchill and Chanel were friends, even close friends. Chanel even attempted to use this friendship during World War II to influence British policy towards the Nazis, with whom she was a close collaborator. But if this question is meant to imply that her relationship with Churchill went any further in personal terms, then I think it's wide of the mark.
According to the Mental Floss website, "Chanel had well–placed friends everywhere, including politicians. She met Winston Churchill in the mid–1920s through her then–lover, the Duke of Westminster. The duke – one of wealthiest men in the world and one with considerable influence – was close friends with Churchill (who was then Chancellor of the Exchequer), and the future prime minister was a regular at his home. Once, in a letter home, Churchill wrote that 'the famous [Coco Chanel] turned up and I took great fancy to her – a most capable and agreeable woman ... She hunted vigorously all day, motored to Paris after dinner, and today is engaged in passing and improving dresses on endless streams of mannequins. ... She does it all with her own fingers, pinning, cutting, looping. Some have to be altered ten times.' More than a decade later, during World War II, this old friendship was used by the Nazis to try to form an alliance with England.
Mental Floss also describes how "After Chanel's death in 1971, classified documents started to emerge that revealed the full extent of her dealings with the Nazis during WWII. Her decade–long affair with Hans Günther Von Dincklage, a German intelligence officer, was well known (she stayed ensconced at the Ritz during much of the Nazi occupation of Paris), but in his 2011 book Sleeping with the Enemy, journalist Hal Vaughan revealed that Chanel was involved enough with the Nazi agenda that she was referred to as Abwehr Agent F–7124 – codename Westminster. 'There were legions of women of courage and derring–do throughout Europe, working hard to outwit the Nazis,' the Washington Post's book review stated. '[But] Chanel was not among them.'"
Wikipedia agrees: "In late 2014, French intelligence agencies declassified and released documents confirming Coco Chanel's role with Germany in World War II. Working as a spy, Chanel was directly involved in a plan for the Third Reich to take control of Madrid ... In September, 1944, Chanel was interrogated by the Free French Purge Committee, the épuration. The committee had no documented evidence of her collaborative activities and was obliged to release her. According to Chanel's grand–niece, Gabrielle Palasse Labrunie, when Chanel returned home she said, 'Churchill had me freed'.
"The extent of Churchill's intervention for Chanel after the war became a subject of gossip and speculation. Some historians claimed that people worried that, if Chanel were forced to testify about her own activities at trial, she would expose the pro–Nazi sympathies and activities of certain top–level British officials, members of the society elite and the royal family. Vaughan writes that some claim that Churchill instructed Duff Cooper, British ambassador to the French provisional government, to protect Chanel."
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