Bridges is widely acknowledged to be ahead, although he is short of guaranteed support to get the 29 required in the vote on Tuesday to replace outgoing leader Bill English.
With support from Bennett - and a group close to her in various camps – Bridges would probably have the numbers and Paula Bennett would almost certainly be elected his deputy.
But Adams', Mitchell's and Joyce's supporters, if they combined, may have enough to stop him.
However if Adams were leader, it would be harder for Bennett to be elected deputy because the party is said to be not ready for two women at the top.
Complicating the lobbying is the fact that several MPs are promoting Judith Collins as Bridges' deputy in recognition of the support she has in the wider party and with the public.
None of the candidates is conceding that they could end up being deputy leader.
Amy Adams would not say whether she was meeting any rivals at the weekend.
She just wanted to have as many detailed conversations with caucus colleagues as she could.
"I haven't done any deals with anyone and as far as I am concerned, this is going to a vote."
Joyce disputes suggestions that he has very little support but he does not rule out the possibility of meeting rivals over the weekend.
"No plans yet but that could occur. Open mind."
He said people had their own motivations for suggesting he had very little support "but I'm happy that my support is strong support and growing but there is work to do."
He said all the candidates were keeping in touch "to keep the lines of communication open because on Tuesday, one way or another, we have all got to work together."
Bridges said he believed the contest would still go to a vote on Tuesday for leader and deputy leader.
He would not be recommending a deputy leader to the caucus if elected leader.
"Ultimately it is for the caucus but I can certainly work with Paula Bennett. I think she has still got a lot to give."
Asked if he could work with Collins as deputy, he said he could work with a range of colleagues.
"But Paula has done a very good job and she has got great experience and I could definitely work with her."