Edwards made the switch to the CRT squad this year after seven seasons on Yamahas - initially with the factory team before joining the satellite Tech 3 operation in 2008.
The 38-year-old American is only fifth among the CRT riders in the championship at present, following a difficult year on Forward's Suter-BMW.
The team is already looking at alternative technical packages for next year and Edwards said he was very keen to stay on and help Forward progress.
"All the staff on my side have been awesome. It's been a pleasure all year and I'm looking forward to next year," he said. "Obviously we've had an issue with some of the package."
Forward is leaving the door open to adding a second MotoGP rider alongside Edwards next year.
It also confirmed that Alex de Angelis would remain in its Moto2 line-up in 2013 but that it could bring him back into MotoGP in '14.