The Government will ask for a long Brexit extension from the European Union if MPs refuse a third time to back the Brexit deal Mrs May struck with the bloc.
Mr Hammond told BBC One's Andrew Marr Show: "We will only bring the deal back if we are confident that enough of our colleagues and the DUP are prepared to support it so that we can get it through Parliament".
She praised the Prime Minister's "dedication" but, when asked if she should resign, she added: 'I think that what is best for her is, really only she knows.
"She needs a dignified departure so however that works best for her for all of the reasons I've said".
And she made it clear she was ready to throw her hat into the ring when Mrs May indicates she is stepping down.
"She has got to look at the fact - when collective responsibility collapses like that, who is the voice of Government?"
Many dead in New Zealand shooting at two mosques during Friday prayers
She said the victims were people who had made New Zealand their home, and that the perpetrators had no place in the country . Four people are in custody and one is person is on the loose after fatal shootings at two Christchurch's mosques today.
Meanwhile, Tory MP Charlie Elphicke warned explicitly that his support for the deal depended on "a change of leadership".
But when the LBC presenter said that Labour leader Jeremy Corbyn has been more focussed on holding a general election, Mr Bradshaw said he doubted its possibility given the "situation we find ourselves in".
The chairman of Ipswich Conservative Association, John Howard, said people were now looking forward to a time after Mrs May had left office: "Everyone is now preparing for the time when there is a new person at the top, and I suspect that includes Theresa May".
He also said that voting for Brexit could unite the UK. And several Tory MPs have told the Daily Mail privately that a resignation promise could be a "game changer".
Marr himself said May was "grinding the opponents of her deal harder and harder", adding: "They are falling one by one by one, aren't they?" It is not enough for the whips to hint that she might quit - people need to hear it, in public, from her own mouth'.
The head of the Vote Leave campaign has urged MPs to swing behind Theresa May's withdrawal agreement, saying Brexit has "never been at greater risk".