Glenn insisted Southgate's next two results in temporary charge of the senior team will have little bearing on whether he gets the job full-time.
Sir Alex Ferguson says Scotland's World Cup qualifier against England at Wembley next month is a must-win game.
Rangnick was linked with the Three Lions post following Sam Allardyce's exit, but Rangnick said talks with the FA only took place during the summer.
Howe, 38, has been mooted as a permanent successor to Sam Allardyce after helping Bournemouth from League Two to the Premier League over two spells.
It was been reported that England under-21s hope to take Rashford, 18, to next summer's tournament, despite him winning five caps for the senior team.
"I've read that Twitter, stories in newspapers, Florentino's puppet at OK Diario kicked me out, but this is a decision I took myself," Pique said.