Ron Mark is rumoured to be making a bid for the role. He has not returned a request for comment.
Ms Martin would not comment at Parliament this afternoon.
Mr Peters was asked repeatedly by media who his deputy leader is and whether he could confirm Ms Martin would remain in that position.
"I cannot discuss caucus business, that's confidential. I am restricted by that. We do not discuss caucus business outside of caucus," Mr Peters replied.
During this afternoon's question time, Mr Peters faced jeers of, "Who's deputy?". Ms Martin was sitting in the deputy chair.
A former Rodney Local Board member, Ms Martin entered Parliament after the 2011 election as No 2 on the New Zealand First list, behind leader Winston Peters.
Her mother, Anne Martin, was elected president in October 2013 after serving as party secretary for six years, and the two are part of a five-strong panel that decides the party list.
That influence has caused tension with some in the party.
Former North Shore City mayor Andrew Williams left New Zealand First after being dropped altogether from the party list before last year's election.
He had publicly complained about his demotion in the draft list from three to 13, and said Ms Martin was threatened by him.
Mr Mark was ranked ninth on the New Zealand First list at the last election.
He was a list MP from the 1996 election until NZ First failed to retain any seats in Parliament in the 2008 election.
Mr Mark was elected mayor of Carterton in 2010 and retained the position in the 2013 election.
- By Nicholas Jones of the New Zealand Herald
