English refused to express any preference for either Bennett or Bridges, saying they were both "excellent" candidates for the role.
After declaring their bids, the two contenders set about competing for his endorsement.
Bennett said her relationship with English was "fantastic" and described him as "awe-inspiring". The two work together on finance and social housing, and they are part of the top tier of ministers, called the "Kitchen Cabinet". But English has also worked closely with Bridges on Auckland's transport problems.
In a sign of a potentially tense contest, Bridges said he would make a better deputy than Bennett because he marked a complete generational change. Bridges came into Parliament three years later than Bennett, in 2008.
Bridges also appeared to take aim at Bennett's lofty role in the Kitchen Cabinet, suggesting she might be out of touch with the party's backbenchers. He, on the other hand, came from outside National's "inner sanctum" and would champion junior MPs.
Bennett shot back, saying she also offered "something a bit different" from the existing regime and would bring energy and enthusiasm.
Meanwhile, four National backbenchers -- Mark Mitchell, Alfred Ngaro, Todd Muller and Chris Bishop -- are leading the calls for a leadership contest. All four are on the cusp of ministerial positions, and are pushing for a contested leadership to secure portfolios in a reshuffle by the eventual winner.
Despite no public endorsements, Coleman was not giving up hope, saying the race was only two days old and would "move up and down" until Monday.
"We're on Wednesday now, the PM resigned on Monday. So there's still a very long way to go."
Collins said she had gained some initial support from colleagues. She was in the race to win it, she said.
"It is a serious bid -- I wouldn't bother otherwise."
The MPs supporting English so far are Bennett, Bridges, Key, Hekia Parata, Murray McCully, Nick Smith, Anne Tolley, Nikki Kaye, Michael Woodhouse, Louise Upston, Nathan Guy, Chester Borrows, Paul Foster-Bell, Nuk Korako and Jacqui Dean.