Lamont has been playing for Rotherham's Under-21 team.
Glasgow are also believed to be considering approaches to three more Scotland midfielders - James McLaren, Ben Hinshelwood and Andy Craig. Hinshelwood has admitted that his future will depend on whether Worcester find a back door into the English Premiership, despite Rotherham's entry to the top flight being rubber-stamped yesterday, and if Bath and Bristol merge.
Meanwhile, Scotland prop Mattie Stewart has left Northampton and will take over as player-coach at first division Bedford. Stewart had been linked with a move to Glasgow.