Both Thompson and Gerrard have lifted the European Cup as Liverpool captains and both have played their part in leading the national team.
"I think Stevie should be given the captaincy. Without doubt," Thompson told Liverpoolfc.tv. "It's nothing against Scott Parker because he's a fine player, a good player, but he doesn't have the pedigree that Gerrard has.
"I don't remember Parker playing in Champions League finals, FA Cup finals and in huge games like Gerrard has over the years. Also there's the number of caps that Stevie has. Everything should point to him being named as captain.
"I would like to think Roy Hodgson will look at it very differently to Stuart Pearce and I would think and hope that Gerrard will get that choice today. He thoroughly deserves to lead England out at the Euros."