The 25-year-old missed out on a call-up for the Three Lions' friendly international matches at Wembley against Germany on Friday and Brazil next Tuesday respectively.
And, while Southgate revealed that he remains a big fan of Wilshere, who has 34 international appearances for his country to date, he suggested that Gunners' boss Arsene Wenger's decision to play him as a number 10 has scuppered his immediate chances of being selected.
On Wilshere, who also played in a similar role when featuring in 27 games during a season-long loan with Bournemouth last season, Southgate said: "That is not the way we are playing. When we have played with a 10, we have had Dele [Alli] and [Adam] Lallana, but we are not playing with a 10 on Friday.
"I don't think Jack is a 10. I think he is a deeper player, but he is not playing deeper. You'd have to ask Arsene why that was.
"But I want to be clear - I really like Jack Wilshere as a player. I think he is a talent and I really respect him as an individual. And I'm hoping that we are in a position in March where we can pick him."