Swansea boss Michael Laudrup says it is likely he will bring in a goalkeeper in January as Michel Vorm needs knee surgery.
Manager Jose Mourinho believes Ashley Cole will earn a contract extension at Chelsea.
The Spanish striker only arrived at Hillsborough last summer from Barcelona B.
Rodri made 11 appearances for Wednesday, scoring one goal during his spell at the club.
A statement from the Championship club read: "Sheffield Wednesday would like to thank Rodri for his services and wish him well for the future."
Meanwhile, Wednesday have reached an agreement with Manchester City over the loan of winger Jeremy Helan for the remainder of the campaign.
A statement from City read: "Jeremy Helan will join Sheffield Wednesday on loan until the end of the season."