Southgate signed a four-year deal in November last year, helping England comfortably qualify for Russia.
The 47-year-old appears set to stay in the role regardless of results at the showpiece event, with Glenn saying he was “realistic” about the nation’s chances.
“Gareth has a long-term contract. He’s not looking at Russia as a free pass,” he told UK newspapers.
“The World Cup’s a really important staging post for our development, I think. We go out there wanting to win it but also being realistic.
“No-one wants to have a bad World Cup but we’re confident he’s the right guy to take us through for the next few tournaments.”
— England (@England) December 1, 2017
England, who won their only World Cup in 1966, will be favoured to progress from Group G after drawing Belgium, Panama and Tunisia.
Glenn said England, coming off Under-17 and Under-20 World Cup wins this year, were learning from the likes of Spain and Germany.
“What we’re doing is making sure that players that have made it through to the first team by 21, 22, have had lots and lots of tournament experience,” he said.
“So, those under-17 winners and under-20 winners will be coming through – and they are serious, serious competitions to win.
“We know that worked for the Germans and we know that worked for the Spanish. So, we’re just repeating things they’ve done with our own emphasis on it.”