Bennett sought to underline her "fantastic" relationship with English.
"I do see myself working really, really well with him ... He has an integrity in his core and a belief in this country that quite frankly I find awe-inspiring.
"We say the same thing, though we might say it a bit differently. He would say I say it louder and quicker."
She said her time in the Kitchen Cabinet had given her a "richness of experience" and that she would bring energy and enthusiasm to the deputy role.
Bridges, however, appeared to say those in the Kitchen Cabinet might be out of touch with the party's back benches, who will play a crucial role in electing the new leadership team. While not referring to Bennett by name, Bridges said he came from outside National's "inner sanctum" and that he would champion the back benches.
"They want to see after eight years of Government that we're listening. And I'm pledging to be that person who will strongly listen to them and be a champion for the back benches."
In declaring his bid, Bridges said he would bring "change and rejuvenation" to the leadership while English would bring the experience and stability. The "whole premise" of his bid was to lead alongside English and he had not considering deputising for any other leadership contender.
"I don't have a Plan B. At the moment, I don't see myself being deputy leader in other scenarios."
Bridges said he would be a better deputy than Bennett because he would mark a complete generational change - though there is only seven years between the two ministers.
In response, Bennett said she also "offered something a bit different" from the status quo.
Bennett entered Parliament on the list 2005, and later won the Waitakere seat before changing to the Upper Harbour electorate. She held the Social Development role for two terms before taking on the newly-created Social Housing portfolio and the Climate Change role.
Bridges became an MP in 2008, beating New Zealand First leader Winston Peters in Tauranga by 11,000 votes. He has held the consumer affairs, energy, and transport portfolios.