Both countries are looking to secure the 21-year-old, who has played 15 times for the Manchester United first-team this season.
McTominary was born in England but qualifies for Scotland through his Glaswegian father.
According to The Times, McLeish travelled to United’s Carrington training ground on Thursday in an attempt to persuade the player to pledge his international allegiance north of the border.
The national team head coach is keen to pick McTominay for the friendly double-header against Costa Rica and Hungary later this month.
It is believed that Southgate will not assure McTominay of a place in his next squad. He will instead look to convince the player that he is good enough to play for England in the future and should resist Scotland’s advances in order to do so.