The former Hazelwick and Pound Hill Junior pupil led the Three Lions to the European Championship final against Italy on Sunday - England’s first appearance in a major tournament final since their famous win over old foes Germany in the 1966 FIFA World Cup final.
But their fairytale run came to a heartbreaking end as England were beaten 3-2 on penalties by the Italians.
The shootout followed a 1-1 draw after 90 minutes and extra-time The Three Lions had taken an early lead through Luke Shaw before Leonardo Bonucci poked home an equaliser in the second half.
Southgate also guided England to the 2018 FIFA World Cup semi-finals and to a third-place finish in the 2019 UEFA Nations League.
A petition for a statue of the England manager in the town was started in 2018 by Spotted: Crawley' s Daniel Armstrong.
And MP for Crawley Henry Smith tweeted before Sunday’s UEFA Euro 2020 final: "We already have a Southgate neighbourhood, but should the new Town Hall be named after Gareth - our proud son of #Crawley - or should their be a statue instead?!"
Cllr Lamb said Crawley Borough Council were reviewing their options on how to best honour Southgate, and that the town should be immensely proud of the England manager for his work on and off the pitch.
He said: “It's clear from the calls for recognition just how proud Crawley is of Gareth Southgate and I don't think it's just due to the fact he's done something no manager has in 55 years, it's the way he did it.
“He's built a team who are not only great footballers, but who come across as genuinely decent human beings and fantastic ambassadors for the sport.
"I don't think there's any doubt we have to do something to honour Gareth Southgate in Crawley, to be honest it's well overdue.
"I'm tasking a cabinet sub-group to review the options and make a recommendation on how best to show just what he means to Crawley.”