But Gers bosses gave the Gunners permission to invite Gilmour and his family to London last weekend, according to The Sun.
Midfielder Gilmour watched an Under-16 game and met Arsenal’s academy head of scouting Steve Morrow.
Interested clubs will be unable to launch a move for the starlet until he turns 16 on June 11.
He could then be pinched for as little as £100,000.
Gilmour was recently invited to train with Mark Warburton’s first team.
He said: “Great experience training with the first team.”
Warburton has urged Gilmour to stay at Ibrox.
“Billy is a talented player and I’m sure many clubs are looking at him,” the Gers boss said.
“Our job is to show him the pathway there is to get to the first team.”