The Take That singer has been on a string of dates with the British actress after first meeting her on the set of a comedy sketch for charity telethon Comic Relief in March - where Catherine played a version of him in spoof band Fake That.
A source told The Sun newspaper: "Catherine and Jason hadn`t met before they were thrown together at the filming.
"But they had great fun as she tried to take him off and he was very impressed.
"There were real sparks between the two."
During filming of the scene Jason apparently remarked that "she`s got great legs" and that "she looked stunning".
The source told the Daily Mirror newspaper: "Jason seemed very taken by Catherine during filming. She was great fun and really got into the spirit of it. They obviously hit it off."
Catherine, 43, and Jason, 41, are said to have been dates including an overnight trip to see a Shakespeare play in Stratford - the playwright`s birthplace - together.
They also both turned out to support comedian David Walliams during his Sport Relief challenge, where he swam the length of the river Thames last month.
Yesterday (19.10.11) Catherine took her daughter Erin, eight - from a previous relationship - to school, but refused to make any comment on her relationship with Jason.
A spokesman for Take That has also said: "We don`t comment on the band`s private lives."