The former rugby league convert was seconded from the English champions by then interim head-coach Stuart Lancaster and helped England secure a second-place finish.
After being confirmed as Martin Johnson's permanent successor, Lancaster made it clear one of his first priorities was installing Farrell alongside forwards coach Graham Rowntree as his lieutenant.
But Lancaster will have to look elsewhere after Farrell decided to remain with Saracens, where is employed as first-team coach.
Farrell said: "The job isn't anywhere near half done [at Saracens] and I have decided I want to help finish the job.
"It has been a special privilege for me to be involved in the Saracens coaching staff for the past two-and-half seasons.
"We have made decent progress during this period but, in truth, as a club, we have barely scratched the surface of our potential."
It means the RFU will have to turn their attentions elsewhere to employ an attack coach to succeed Brian Smith.
Despite Farrell's popularity with Lancaster and the squad, England only mustered six tries in the RBS 6 Nations and the lowest number of line breaks by any team.
New Zealand's World Cup winning coach Wayne Smith is thought to be one of the prime candidates to fill the position and RFU chief executive Ian Ritchie confirmed the hunt will begin in earnest.
He said: "While Andy was clearly an important part of the coaching team during the course of the Six Nations, we fully respect his wishes to continue as a coach at Saracens.
"We will continue to work hard with Stuart Lancaster to build a world-class coaching team as we look towards the Rugby World Cup in 2015."