The Sun says Rodgers, 36, managed just five league wins in his Madejski Stadium reign after taking over in June.
The dismal record leaves the Royals in 21st place in the Championship.
But owner John Madejski is now turning his sights to Rodgers' replacement.
And ex-Middlesbrough boss Southgate is believed to have shot straight to the top of his wish-list.
Royals chief scout Brian McDermott takes over as caretaker, while backroom staff Dean Austin and Frank Lampard Snr depart.
One Reading first-teamer admitted: "I had no idea it was coming."