The ex-Liverpool captain confirmed on Thursday that he will be hanging up his boots after 18 years as a professional footballer and revealed he has already down the chance to take charge at League one side MK Dons.
Although Reds boss Jurgen Klopp has made it clear that the Anfield door will always remain open to Gerrard, Thompson believes the former LA Galaxy should learn his trade under England U21s boss Aidy Boothroyd before taking on any significant managerial roles.
Thompson, who worked with Gerrard during his time as Liverpool's assistant manager, told Sky Sports: "There will be a lot of opportunities but I'd rather see him with England U21s, which would probably be my preference, then come back to Liverpool in some capacity, maybe with the U23s or as first-team coach.
"Aidy Boothroyd is down there in an interim role with the England U21s with a wealth of experience. He's gone into the under-21 setup [and] is the sort of figure to learn from.
"It's one thing cutting your teeth with the younger players at a football club but going into a League One or League Two side is a difficult scenario unless you've got a senior coach with you."
He also said it was unlikely Klopp would want him to work with the Liverpool first team at present, explaining: "With the team doing to so well it could take the focus away, especially with Jurgen Klopp's trusted lieutenants doing such a great job."
After speaking to Gerrard “a couple of months ago”, Thompson admitted that he had expected the midfielder to continue playing in some capacity.
"He was still unsure about where his next step was. The one thing he was sure about was leaving LA Galaxy," he said.
"He was hoping to be a player-coach somewhere, which would have given him some coaching experience as he's not got any whatsoever.
"You just felt with the way he retired quite suddenly, having known he considered carrying on player and coach, that maybe he has something in the pipelines."