Southgate, 46, was placed in interim charge after Sam Allardyce left the role in September and has overseen wins over Malta and Scotland and draws with Slovenia and Spain.
He has steadfastly refused to publicly declare an interest in taking on the role full-time, insisting he was focusing purely on the task in hand.
But Glenn told Sky Sports News: "He's made it clear over the last week that he's convinced he wants the job."
Glenn said Southgate would shortly be interviewed for the job.
Asked if the FA would interview any other candidates for the role, he replied: "Potentially."
Southgate stepped up from his role as England Under-21 manager after Allardyce left his position in the wake of a newspaper sting.
"Southgate's in a strong position," Glenn added.
"Because not just the facts of the last four games have shown a lot of signs of encouragement, but the fact he's worked in the FA for the last couple of years."
England's next game is a friendly away to Germany on March 22.