Rovers made a late bid to sign the 29-year-old before Thursday's loan deadline and although the club's administrator Brendan Guilfoyle accepted the offer, Fallon immediately turned down the approach.
Reid told the Plymouth Herald: "I'd been told on the phone by the administrator that we needed to take up the offer from Bristol Rovers to take Rory on loan as the club needed the money.
"I'm fully aware of the situation and know the administrator has his job to do and I told him 'well, it's up to the player'. Not long afterwards, I had a call from Rory who said basically there was no way he was going anywhere while the club was in the position it was in.
"He told me 'listen, we've got eight matches left this season and I want to be involved and fight for points alongside the lads - I'm not going anywhere while the club is in this kind of trouble'.
"Rory's attitude has been brilliant throughout and his attitude simply reflects the great spirit we have here at Plymouth Argyle. He's been top class for us and so have all the other lads."
Source: PA
Source: PA