The pair have allegedly been warned by acting controller Ben Cooper that they could lose their slots to fresher and more relevant talent.
Edith Bowman and Trevor Nelson are also among the stars in jeopardy, according to The Sunday Mirror.
"A few of the DJs have grown up at Radio 1 and in some cases have settled down and have families," a source said. "But that means they're not as in tune with the younger generation as they used to be."
A spokesperson for the station itself refused to engage with the report, stating: "Any story about schedule changes would be pure speculation."
Kay joined Radio 1 in 2004, while Cox made her debut in 1999 and held the station's coveted weekday breakfast show slot between 2000 and 2003.
It was reported last month that T4 host Jameela Jamil is in talks to join Radio 1.
Kay's future at the broadcaster was previously in doubt following his sex text scandal last year.