The 'Price Tag' singer is currently appearing as a coach on BBC One's The Voice UK, but Cowell has allegedly put forward a move to bring her to ITV for The X Factor.
"I love that girl. She's one of the best things to come out of British music in years. I want to snap her up," Cowell reportedly said.
Cowell will be willing to rise her wages from £200,000 at the BBC to £600,000 at ITV, according to The People.
"He thinks Jessie is a huge star," a source said. "He wants her and he will do anything to make sure she is his.
"He knows it could be a tough battle but it is one that he is determined to win. Simon is always looking to bring down the age of his judging panel and wants to make the show appeal to a younger audience. She is the perfect person to do that.
"He doesn't care what it takes. Simon wants her and he will do whatever it takes to get her. He rarely doesn't get what he wants."
A BBC insider has responded by saying that Jessie J will not be available to swap shows later this year.
"The BBC are so thrilled and happy with landing her. They absolutely love her," they said.
"She is a brilliant person to draw in a younger audience and the BBC won't take kindly to Simon trying to nick her. She was a brilliant coup for The Voice so Simon had better keep his hands off."
Simon Cowell had previously signed Alesha Dixon for Britain's Got Talent from BBC One's Strictly Come Dancing by raising her salary from £100,000 to £350,000.
"We pinched Alesha," he added. "In the grand scheme of things that's fun, that's not being ruthless. It's just evening the score. The BBC always said they are not there to compete, which is a complete load of rubbish.
"They are the most competitive people I've ever met. I don't dislike them for it but you've got to be honest. They throw everything at Strictly for one reason only - to beat The X Factor - and they are putting The Voice on to compete with Got Talent. It was only right I did something back to them."
Simon Cowell's London home was broken into by a female fan on Saturday night (March 24).
> The Voice and Britain's Got Talent: First review - our verdict