The 32-year-old DJ and the 47-year-old singer have grown close in Las Vegas while working on new music together and their relationship is believed to still be quite new.
A source told the National Enquirer: "Calvin and Jennifer are very careful to not be seen together.
"It's very early days and they're keeping things on the down-low.
"They've become really close. There's epic chemistry between them."
Calvin attended Jennifer's birthday party in Las Vegas in July and shared a number of clips of them together at the bash on Snapchat.
He split up with Taylor Swift, 26, a few months ago, while Jennifer - who has eight-year-old twins Max and Emme with her ex-husband Marc Anthony - and her backing dancer Casper Smart recently ended their on/off relationship.
A source previously explained: "Jennifer and Casper have split. They have not been together for a few weeks. It wasn't anything dramatic and they were on good terms - it just came to a natural end.
"They will remain friends and they see each other, but they are not together. It was very amiable."
Following his split from Taylor, after 15 months of dating, Calvin was linked to Tinashe and Mexican model Eiza González.
While insiders called Tinashe a "good distraction", Calvin's close pal John Newman insisted he didn't need anyone to comfort him after his relationship with Taylor ended.
He said: "He doesn't need anybody. His emotions are so strong, he's so strong minded.
"He's a one-man train who is never going to stop. The limits are literally endless for him.
"What he's done is nothing compared to what he can do. That's the scary thing, how far this really could go for him."