Graham, a boyhood Ton fan, was recently released by Hamilton Academical after the club were relegated from the SPL.
The defender made 17 appearances for the Lanarkshire side this season, but was cut loose as the club prepares for life in the First Division.
Graham served under Allan Moore at Stirling Albion for five years - and the Ton boss was unsuccessful in a move to bring the player to Cappielow last summer.
But Moore has now returned for the 27-year-old, and is set to snap him up on a contract once he returns from holiday.
The Ton manager has also moved to bolster his attacking options, with Ross County striker Paul Di Giacomo also agreeing to sign.
The 28-year-old has agreed in principle to a switch to Cappielow and is expected to finalise a deal when his current contract expires at the end of the month.
Di Giacomo also played for Moore at Stirling, scoring six goals during a loan spell from SPL side Kilmarnock in the latter half of the 2004/2005 season.
Since then, the hitman has had spells at Partick Thistle, Airdrie United and County, where his time was hampered by a series of niggling injuries.
The Ton attack is in need of reinforcements after Derek Lyle and Stewart Kean were released this month, while Brian Graham has joined Raith Rovers.
And while neither deal is signed and sealed, Moore is hopeful he can tie the pair up on permanent contracts by the end of the month.
Moore told the Tele: "We have spoken to both players and they have agreed to sign, although they are not able to join us yet.
"Paul Di Giacomo's contract at Ross County does not end until later on
this month but he has agreed to
sign for us."
The new arrivals are set to follow former Stirling captain Ross Forsyth in joining Ton - with all three players having worked with Moore before.
And assistant boss Mark McNally is excited about joining up with his former charges at Cappielow.
McNally said: "The three guys will all be signing for us. We have worked with them before and look forward to doing so again."