Sheeran appeared on The Late Late Show on Monday (28 June), telling host James Corden all about his friendship with the Friends actor.
In fact, they're such good mates that there's an ongoing practical joke that he likes to play on her - a joke that involves bondage masks, of all things.
It all goes back to a time when Cox proudly showed off one of her household gadgets to Sheeran, who soon realised he could use the power of modern technology to his advantage.
It was an Amazon Alexa digital assistant, which can be used to order items with a simple voice command.
He told Corden: "She has this Alexa thing in her house, and she goes, 'Ed, isn't this wonderful? This is my Alexa. I can just order whatever on this.'
"So she walks out of the room and I go, 'Alexa, order me a gimp mask.'"
Hilariously, the first mask that Sheeran ordered was initially discovered by Cox's assistant, who discreetly left it out for her boss without saying anything.
The singer explained: "She opened the post and she finds this mask, and instantly goes, 'Oh, I don't think I was meant to see that,' and leaves it on Courteney's bed.
"And Courteney comes up and she's like, 'Where did this come from?'"
Over time, Sheeran's cheeky prank has turned into a bit of a tradition - meaning now she has several bondage masks at her disposal.
Sheeran continued: "Now every time I go back there, I order her another.
"She has maybe, like, 12. I hide them in people's bedrooms. So they'll go in the drawer and they'll just find this leather S&M mask."
Sheeran also showed viewers a special video message Cox once sent him, in which she whipped out a special prop as she wished him a happy birthday.
"Ed, Happy Birthday," Cox says in the clip.
"I just can't believe that you're turning 30!"
A voice off-camera then calls her away as she's needed on set.
"I gotta go," she says.
"I just want to say thank you so much for all the protection. These have really come in handy."
Zipping up the mask, she quips: "No Covid here!"