Wayne Rooney remains the favourite to lead the Three Lions into their Euro 2016 qualifying campaign, though the former West Brom boss has not yet made a final decision
Hodgson must decide who to hand the armband to after Steven Gerrard announced his retirement from international football following England's disappointing World Cup campaign.
Lampard was vice-captain in Brazil despite not starting the first two matches, while Manchester United striker Wayne Rooney is one of the favourites for the role.
England play their first fixture after the World Cup against Norway at Wembley on September 2.
And while Hodgson revealed Lampard may lead England out for that game, he believes the 36-year-old's main focus will be on helping Manchester City during his loan spell before starting his new venture with New York City in 2015.
"Frank is the vice-captain and there is Wayne Rooney," Hodgson told Sky Sports.
Asked if he had chosen his new skipper and whether they had been told, Hodgson replied: "Yes to the first part and no to the second.
"We don't need to compare to other European countries, let's just stay as England. I have got a big job ahead of me I believe to choose that captain and that's what I'm working on.
"I have half-chosen him. The person who is captain for the Norway game will be the person I will be designating to hopefully be the captain in 2016 [at the European Championships in France].
"At the moment, Frank's concentration has to be on this Manchester City job and in New York, where he could very well be captain.
"I don't think I can expect him to say 'please make me captain because in 2018 [at the World Cup] I want to lead the team out', because I don't think he will be thinking that himself.
"But having said that, if he is with us in the first game, who knows. Then he could captain us that day."
The 67-year-old also reflected on England's performance at this year's World Cup, where they were eliminated at the group stages without a win, adding: "[There is] still great disappointment, and the feeling of enormous frustration because the preparations had gone very well."We have a lot of rebuilding to do not only of rebuilding our team but also rebuilding all the faith that people showed us.
"We have to look forward to 2016 and make certain that we don’t allow it to happen again and hope we don't have the same lack of fortune as well that we perhaps suffered as well."