The 31-year-old is a free agent after his loan at Dundee – where he made 17 appearances and scored three goals – came to an end last week.
According to Sky Sports, Falkirk have have moved quickly to sign the former Scotland international, with manager Martin Rennie aiming to guide the club into the Championship. He recently appointed Kenny Miller as his assistant.
Griffiths signed a new one-year deal at Parkhead last summer, but was released by the cinch Premiership leaders after making 261 appearances and scoring 123 goals over eight years.
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After confirming his departure from the club, Celtic boss Ange Postecoglou told Sky Sports News: "He had an outstanding football career at Celtic.
"All of us, we kind of pass-through clubs, but you want to leave a mark there when you do.
"Leigh has certainly done that. I think he can look back on his Celtic career with great pride and he goes on to the next chapter of his football career. We wish him all the best."
Griffiths spent time at Leith Athletic, Inch Colts and Hutchison Vale as a youth and made his senior debut as a 16-year-old at Livingston.
Dundee paid £125,000 to sign him just before he turned 20. He continued to flourish at Dens Park, earning a move to Wolves.
He was loaned out to Hibs, the club he supports, for his first two seasons, 2011/12 and 2012/13, before returning to Molineux when they were in League One. He became top scorer at Wolves before being signed by Celtic in January 2014.