Rumours surfaced on Monday that Lennon was facing the axe in the wake of Saturday's 2-1 Scottish Premiership home defeat to Partick Thistle.
St Mirren have endured a difficult start to the season having picked up just one point in four games and also crashing out of the League Cup to Queen of the South.
Despite the pressure growing on Lennon, Gilmour insists no decision has been made on the manager's future and that he remains in the job at St Mirren Park.
"If people want to talk about the manager, then I have nothing to say," Gilmour told the Daily Record.
"Danny is the manager of St Mirren and getting on with his job. That's it."