The presenter’s name began trending on Twitter almost immediately after Chris announced he would be leaving the station after eight years on the show.
Coxy is already Chris’ regular stand-in on the 6.30am to 9.30am weekday slot, and many have named her as their top pick to take on the role full time when Chris leaves in December.
Sara currently hosts the 10pm to midnight show from Monday to Thursday, after having impressed bosses with her Sound Of The 80s show.
Like Chris, she previously the hosted the Radio 1 Breakfast Show from 2000 to 2003, before being succeeded by Chris Moyles.
However, the bookies are backing someone else for the job, as Zoe Ball is their current favourite, with odds of 4/7 with William Hill.
Sara follows in second place with odds of 6/4, while Simon Mayo has 8/1.
The station is under pressure to increase the number of female presenters on its daytime schedule, as there are currently no women with a permanent show airing between 6.30am and 5pm during the week.
Chris has announced he will be going on to host the Virgin Radio Breakfast Show after leaving Radio 2 at Christmas.
In a statement, he said: “I see nothing but exciting and groundbreaking opportunities ahead. In a medium that is changing so quickly on a daily basis, the potential for growth is unprecedented.
“Our plan is: to give it all we’ve got, see where we can get to and have the most possible fun along the way. It makes me smile every time I think about it.”