The 19-year-old midfielder’s future has come under the scanner since the sacking of Frank Lampard by Chelsea last week.
Thomas Tuchel has tried to calm the nerves of Chelsea fans who feel Gilmour could get lost in the crowd under the new manager but he has not featured in the matchday squad in the first two games under the German.
And there are claims now that he could leave Chelsea on loan for the rest of the season on deadline day.
Southampton have now been linked with an interest in signing Gilmour on a loan deal.
But it is far from certain that the young Scotsman will join the Saints.
Ralph Hasenhuttl is a fan of the Chelsea youngster and appears to want him in his squad for the rest of the season.
A deal could hinge on guarantees of game time and minutes as Gilmour wants to be in the Scotland squad for next summer’s European Championship.
