Weir was appointed as Danny Wilson's permanent replacement in May and will lead his side and he will be eager to make a strong start against the Magpies.
The game at Bramall Lane will also launch the Football League's 125th anniversary celebrations, with County, the world's oldest Football League club, visiting the world's oldest professional football ground.
The Blades were unbeaten in eight summer friendlies, including a recent 2-2 draw with Greek Super League side Atromitos.
And former Everton player and coach Weir told the club's official website: "We have played well throughout pre-season and to compete with a team like Atromitos on a level playing field is testimony to our progress.
"We are progressing well in terms of preparation for Notts County and everyone cannot wait to get started."