Along with his role in Brexit talks, Mr Davis said he had an "invisible job" of preparing the United Kingdom for the possibility of no agreement with the EU.
Some leading members of Britain's ruling Conservative Party want finance minister Philip Hammond to become prime minister, replacing Theresa May and steering the country through the Brexit process, the Sunday Times reported.
Questioned on May's insistence that "no deal is better than a bad deal", Davis admitted the United Kingdom may walk away from negotiations with the European Union if it does not get what it wants. I know she is coming under a lot of pressure at the moment, but I have seen her in action. She's bold. She takes her time.
"Point number two is, I want a stable backdrop to this Brexit negotiation".
Asked about a meeting this week where a fellow MP called for him to be the party's next leader, Mr Davis replied: "I said afterwards our job is to support the Prime Minister and make Brexit work, not anything else". My message to [rebel MPs] is: get on with the day job.
Remainer Ken Clarke said: "If the sillier people in the Conservative Party now decide that the best thing to do is to start arguing about the leadership, then you just make a very hard decision more absurd".
Mrs May was fatally wounded after losing her majority in this month's election and having to rely on the DUP to prop up a minority government.
However, while he said he took his "share of the blame", he suggested that the party's misjudged election strategy was responsible for the result. "I didn't design the campaign".
But he warned Tory MPs not to launch a bid to push out the Prime Minister in fear of destabilising the Brexit process. The PM's brand is so damaged it is painful. She makes good decisions.
Another contender for the leadership is Mr Johnson, but the Foreign Secretary has ruled out running until after Brexit is delivered in 2019, arguing that the Government has to "get on and deliver" a deal.
Mr Davis, Boris Johnson and Philip Hammond are all hotly tipped to swoop into the top job if she is booted out of office.