Those close to both are at pains to promise that there is nothing going on between them. Enquirers are told that they are "just good friends" of long standing, and that it would be wrong to imagine that a romance is going on.
Is this really the case though? Chris, 38, is single despite being at the centre of an entertainingly complex web of flirtations and entanglements.
He is, I'm told, in an "of"' phase of his on/off romance with Oscar-winner Jennifer Lawrence, 24. Kylie, 47, is also single, endlessly unlucky in love, and 5ft nothing of Tinker Bell gorgeousness and self-confidence.
Anyone who saw her perform that night will know she is at the peak of her loveliness, and utterly assured in it.
What few people know, though, is that there is a burning attraction between her and Chris which goes back almost 15 years. Indeed, Kylie had her eye on Chris (and vice-versa) long before he met Gwyneth Paltrow in November 2003.
It is this acknowledged chemistry between them that may be behind a certain amount of knowing chatter about this week's 1.30am stroll in the park. With no official comment on the nature of their relationship such speculation will only flourish.
Kylie and Chris seem to have encountered each other in 2001. They are stablemates at the Parlophone record label and were on the same bill at various festivals. When they met, he, a shy Devon boy, was getting over the end of his first serious romance with radio executive Lily Sobhani.
Chris went to the single sex public school Sherbourne, before studying history at University College London.
He was a virgin until today's almost unheard-of age of 22 and, to say the least, unconfident with women when he became famous. He told one interviewer he respected women too much for casual sexual flings.
Kylie came to a Coldplay concert in 2002 and made sure that she met Chris. She chirruped afterwards: "Chris just turns me to jelly. He's a really sexy singer." The feeling was evidently mutual, and a friendship developed.
When Kylie was diagnosed with breast cancer in 2005, Chris dedicated the song The Scientist to her at a concert in New York. "This song is for Kylie Minogue," he said, "please get better soon."
At Glastonbury later that summer (which Kylie had hoped to play before her illness), he sang a version of her hit song Can't Get You Out Of My Head and told the crowd: "We should remember absent friends."
During Kylie's comeback in 2007, he was among those who sent a video message to the Music Industry Trust awards. Chris, by now married to Gwyneth and with two children, said: "I thought I was over my obsession, but Kylie is pretty amazing. We all fancy her rotten."
Perhaps most significantly, the following year, as talk swirled of a crisis in his marriage, he invited Kylie to join Coldplay at recording sessions in London. She recorded a duet with Chris which he later dropped, saying it had 'too sexy' a vibe for the finished work, their fourth album, Viva La Vida.
Chris said Kylie was in "particularly brilliant form" but added: "It's just too sexy. At this point we cannot be that sexy."
The duet was eventually released as a charity song to raise money for AIDS victims in 2008.
It may just be coincidence, but 2008 and 2009 were said to be "dark days" in the Paltrow/Martin marriage. They were never seen together in public - at his insistence.
Meanwhile, Gwyneth hung out with billionaire socialite and hotel magnate Jeff Soffer in Miami, and Chris was spotted offering pop star Alesha Dixon a ride home in his jet from an event in Cannes.
In 2009, there were even reports (denied) that Chris had been seen kissing an actress friend of the couple, and that he and Gwyneth were to divorce. Legal action was threatened but never materialised.
Gwyneth started taking on all kinds of unusual endorsement deals, including one for bagged salad, and there were suggestions she was doing this in case her marriage did not last.
She appeared to confirm problems when she said in a magazine interview in 2010: "Sometimes it's hard being with someone for a long time. We go through periods that are not rosy."
They separated in March 2014 and the divorce was finalised in April this year.
Since then, Chris's private life has been ever more colourful. Indeed, this slightly balding, milk bottle-pale Devonian, a late starter with women, has surprisingly become a ladykiller at a world-beating level.
First to be linked with him post-Gwynnie was TV presenter and style icon Alexa Chung. They were spotted enjoying a drink together in New York in May 2014.
This followed reports that they had been hanging out together the previous summer at Glastonbury, enjoying being served ice cream at 3am once. Chris denied that he and Chung were an item.
Next came a romance with actress Jennifer Lawrence. Again, they met while he was still married, at the Met Ball in New York in 2013. (Although in her "conscious uncoupling" statement Gwyneth did say they had been separated for periods before the split.)
US Weekly magazine reported that Chris flirted up a storm with Jennifer at the event and felt they had "amazing chemistry." They met again at a party after a Coldplay concert.
Their first appearance as a couple was in February 2014 in Los Angeles at a birthday party for One Direction star Harry Styles. Embarrassingly, Alexa Chung was also there.
It was a taste of the tangles to come, for although he was by now single and ostensibly free to date Lawrence, he was very much a part of Gwyneth's life still.
There were family holidays, and he even went with her on business trips like one to Dallas to promote her well-being website Goop.
They spent Valentine's Day together. He hired a cottage opposite hers in the Hamptons, the upmarket holiday destination outside Manhattan, so that they could be together as a family all summer.
Last October, he and Gwyneth had a dinner (just the two of them) at the restaurant Katsuya in Los Angeles. As they were saying their goodbyes, Gwyneth leaned out of her car and stroked his face in full view of the waiting paparazzi.
This, according to multiple published reports, was too much for Jennifer, who promptly dumped him.
They got back together, with Jennifer's grandmother saying that he had come to meet the actress's mother and might even be around for Christmas.
However in March this year, the circus of competing blonde sexpots in pursuit of Mr Coldplay kicked off again when he went on a beach date with actress Kate Hudson, the stunning daughter of Goldie Hawn.
Gwyneth, who introduced them a decade previously, was in Singapore on Goop business.
She was apparently annoyed because she and Chris used to go out as a four with Kate and her then boyfriend Matt Bellamy, of the band Muse.
And Jennifer Lawrence? Friends of Chris swore that it was all completely innocent, and pointed out that he and Kate Hudson were 'work-out partners' at the same gym in LA.
Maybe that contributed to instability between Chris and Jennifer. One friend of the singer tells me: "They have been on and off a lot."
It could simply be, however, that having exited a long marriage, he is not in the market for a serious relationship. He and Jennifer spent Easter together in New York, sharing brunch at Central Park's Tavern On The Green before visiting the John Lennon memorial, Strawberry Fields.
They were last seen together at a charity event in May but they seem to have split again. A source says: "Chris is at last taking advantage of his hottie status. All the way through his marriage he couldn't be the rock star sowing his wild oats.
"Both he and Gwyneth had a lot of therapy, as she admitted, to try to come to terms with having a shotgun marriage when she was pregnant with Apple, at a point where he was really young and still inexperienced.
"Although she has a new boyfriend, [TV producer and director] Brad Falchuck, Gwyneth clearly likes to have Chris in her life a lot.
"She says that they now have the relationship which they always ought to have had and is proud that they have worked out a really close friendship.
"There is a school of thought that she has been telling Jennifer Lawrence to back off by posting pictures of him on Father's Day and so on.
"They are very much in each other's pockets still and she polished him up like a diamond; she taught him how to dress, how to cut his hair, what clothes to wear. They remain very, very close."
But not, perhaps, close enough to stop him trying to woo Kylie, the girl of his dreams so long ago.
- Daily Mail
