The FA are of a mind that former central defender Southgate, 47, can continue to oversee England's rise to become a true world force, inside the top four countries in the sport.
"I think we'd both like that, but if we talked about it at any length it would then be a contract conversation and he's on holiday now - so we'll talk when he comes back," said Glenn.
He’s really blossomed. He fits the definition of the modern manager we want. It’s not just picking the team, but all the aspects behind it. We’d like him to stay beyond 2020.
England manager Gareth Southgate (right) and captain Harry Kane (left).
Image credit: PA Sport
Southgate signed a four-year contract in November 2016 having initially landed the job as a caretaker manager.
His stock soared in Russia, and he has been nominated on the 11-man shortlist for FIFA's manager of the year as England fell in love with their national sport all over again.
Southgate could be tempted by a return to club management after being dismissed by Middlesbrough in 2009, but he is likely to be given an increase to the £1.8m-a-year salary he agreed two years ago.
"I want to pay my staff competitively so they are not a flight risk, but I also recognise we are a sports governing body and we're not a Manchester United," said Glenn.
We can never compete with the Premier League in terms of pay - everyone knows that.
"At some point, Gareth may choose that he wants to try something different and we'll find someone else."
Glenn believes the run to the last four of the World Cup is only the start for Southgate's England. They enjoyed wins against Tunisia and Panama, Colombia on penalties and a victory over Sweden to reach the semi-finals.
Gareth Southgate speaks to FA chief executive, Martin Glenn during the The Emirates FA Cup Semi Final match between Chelsea and Southampton at Wembley Stadium on April 22, 2018 in London.
Image credit: Eurosport
A couple of defeats to Belgium, one in a group stage dead rubber, and the third place play-off match, has not dissuaded the FA that their coach is on the right track.
"Gareth is on a journey," he said. "He loved the World Cup and he's built his own belief that we can go further and that motivates him and (assistant) Steve Holland.
One of the most powerful things he said after our defeat by Croatia was we haven't done the job. We want to be a top 4 team, one of the best, so it's unfinished business.