Barnes played 14 minutes for England in a friendly against Wales in October 2020, but has not featured under Southgate since, with rumours multiplying he is considering switching allegiance to Scotland through his maternal grandparents.
In an unusual quirk, Newcastle United team-mate and Whitley Bay-born Anderson trained for two days with the current Scotland squad last week before withdrawing with injury, but could pledge his international future to England.
"They are both very good players," Southgate said before England meet Scotland in a friendly at Hampden Park on Tuesday.
"Harvey is a player we are always monitoring and someone we like a lot. There is a lot of competition in that area of the pitch. Elliot is a player who has progressed really well and Newcastle rate him very highly.
"Sometimes you cannot offer a player something as quickly as they would like and I do not know the answer to the ultimate question for them."
Meanwhile, Scotland boss Steve Clarke will take stock of their respective situations after the friendly against England.
He said: "We want the best possible players we can get, if they are eligible for Scotland and they have a chance to play for us and they can improve the squad - which is not an easy thing to do.
"I get all the story around it because we are playing England. But I quite like speaking about the boys I have got because they have put us in a really good position."