The Sun says Southampton could make a January move for Blackburn striker Rhodes, 24.
Saints boss Ronald Koeman has admitted he is in the market for a new striker signing.
He said: "I don't think that we need a defender, I don't think that we need a midfielder.
"It is only up-front we do not have a lot of options. I was waiting to have Jay Rodriguez back in January but his set-back means now that we need another one.
"That makes it a little bit more difficult for us and that's the reason that we are looking to bring in another attacker."