The 21-year-old has netted twice for Frank Lampard's side across their opening three games of the new Premier League campaign and has proved himself to be a potent threat off the ball as much as on it.
The Blues finally recorded their first win under their new manager when they edged to a thrilling 3-2 victory against newly-promoted Norwich at the weekend, with their young striker scoring both their opener and a sublime winner either side of Mason Mount's effort.
Abraham was previously called up by Southgate in November 2017 while on loan with Swansea, making two appearances for the national side, before dropping back down to Under-21 level, where he was a member of the Toulon Tournament-winning side the following year.
Now Sutton, who spent a season at Stamford Bridge between title-winning spells with Blackburn Rovers and Celtic, has backed the forward to force his way into the Three Lions manager's thinking ahead of a vital stretch of games that sees them face Bulgaria, Kosovo, the Czech Republic and Montenegro.
“I thought Chelsea were very good," Sutton told BBC Radio 5 Live. "You talk about England and I think the interesting one is Tammy Abraham. I know it’s early days but I thought he was very good.
"When you think of England strikers you’d say shoo-in Rashford and Kane but then you start to think. Could Abraham push Callum Wilson?
"Yeah, I think he could. Frank Lampard seems to have great faith in Abraham, getting the nod over Giroud.
“I was impressed with Abraham at the weekend. It’s a big season for him and the fact that he has Frank’s backing, good mobility and no doubt he took both goals fantastically.”
Southgate named three strikers in his squad for the Nations League finals earlier this year, with Kane and Rashford joined by Bournemouth forward Wilson.
Chelsea, meanwhile, take on another new face in the Premier League this weekend when they welcome Sheffield United on August 31.