Gordon has starred for England at youth level and was player of the tournament as the Under-21s won the European Championship in the summer.
The 22-year-old has not been capped by the senior side yet, though, and reports claim Scotland now want to call him up as Gordon qualifies for them due to his paternal grandparents.
Southgate, however, has told the forward not to rush into any decisions because he is close to breaking into the senior squad.
“He is a good player. He was very good with our under-21s last year. He is very close, I have to say,” said Southgate.
“I like his attitude, I like the way he works so hard for the team. Of course, I am sure he was hoping for a call at this moment, but there are good players playing well in that position on the field.
“But I think, like a couple of others, they have always got to think a little bit further ahead as to opportunities that can come up.
“I think for somebody like Anthony that is eminently possible with us.”
Scotland are reportedly looking at a number of players who have been capped by England at youth level but could switch allegiances.
As well as Gordon, they include Elliot Anderson, Harvey Barnes and Tino Livramento.
Scotland assistant John Carver said: “If someone’s got a relation who is Scottish, we have to look at it.
“It’s then down to what he decides, but we are aware that these are big decisions about their future. It’s no secret that Newcastle are my team and I watch them more than anybody.
“We are aware they have several outstanding players, who have never been capped by England at senior level, who qualify for Scotland.”