Chelsea's Gary Cahill and Liverpool's Jordan Henderson are also believed to be under consideration, but Southgate has indicated that England could enter next summer's tournament without a permanent skipper.
"There have been times where having one leader is important. But I feel as if the modern world is a little bit different and the shared responsibility becomes a more important thing," Southgate told reporters.
"I feel the process has been really revealing for us as a group of staff to watch, but also a good experience for the players to feel that responsibility and to share the ownership.
"Too much has fallen on, in particular, Wayne's shoulders in the last few years. And now there's been opportunity, even in meetings and on the training pitch, for others to step forward and make the contributions and give an opinion, which I think is really important. They're the guys on the field that have to make decisions."
England will take on Belgium, Tunisia and Panama in the group stages of the 2018 World Cup.