He said a no-deal Brexit, something Mr Johnson has left open as an option, was “not something I could ever sign up to”.
Asked if he thought he would be sacked next week, Mr Hammond said he would resign on Wednesday to Theresa May.
He said he intends to quit after Prime Minister’s Questions but before Mrs May steps down.
Speaking on the BBC’s Andrew Marr Show, Mr Hammond said it was important the next PM and his chancellor were “closely aligned” on Brexit policy.
Mr Johnson has said the UK must leave the EU by the new Brexit deadline of 31 October “do or die, come what may”.
His leadership rival Jeremy Hunt has said a no-deal exit cannot be ruled out, but he is prepared to further delay Brexit if required to get a new withdrawal deal.
Mr Hammond said the situation “might be more complicated” if Mr Hunt wins the Tory leadership contest, but “all the polling” suggested Mr Johnson would succeed.
“That is what is likely to happen, and I’m making my plans accordingly”, he said, adding he would wait until the result is announced on Tuesday to “see for sure”.
Mr Hammond said he understood committing to leave by this date, even with no deal, would be a condition for serving in Mr Johnson’s cabinet.
He said: “That is not something I could ever sign up to. It’s very important that a prime minister is able to have a chancellor that is closely aligned with him in terms of policy”.
He added that Jeremy Hunt’s position regarding a no-deal Brexit was “more nuanced”, and he had not demanded a “loyalty pledge” on the exit date from prospective ministers.
Mr Hammond said he would support either man in their pursuit of a new Brexit deal, but it would not be possible to agree this before the end of October.
“A genuine pursuit of a deal will require a little longer”, he added.
Mr Hammond has been a prominent critic of the idea of a no-deal Brexit, recently indicating he may vote to bring down the next PM to stop such a scenario.
He had said he could “not exclude anything” when asked whether he would back a motion of no-confidence in the government.
Asked whether he would go against the next PM in a vote of no confidence, he said: “I don’t think it will get to that”.
“I am confident that Parliament does have a way of preventing a no-deal exit on October 31 without parliamentary consent”.
“I intend to work with others to ensure Parliament uses its power to make sure that the new government can’t do that”, he added.
Earlier, Justice Secretary David Gauke reiterated his intention to resign from government should the next prime minister pursue a no-deal Brexit.
Mr Gauke told the Sunday Times: “If the test of loyalty to stay in the cabinet is a commitment to support no-deal on October 31 – which, to be fair to him, Boris has consistently said – then that’s not something I’m prepared to sign up to.”
The votes haven’t been counted – but already Westminster is preparing for Prime Minister Johnson.
It’s not a surprise that Philip Hammond has decided not to serve in a Johnson government.
But the manner of the announcement – live on television, hammering Mr Johnson’s key policy on Brexit so publicly – shows just how deep divisions in the Tory Party run.
Justice Secretary David Gauke has confirmed he’ll quit too if Mr Johnson wins – and others are likely to follow.
There is an element of jumping before they are pushed.
But it’s also a reminder the next PM will face the same huge challenge Theresa May faced – how do you manage discipline in a bitterly divided party, with such a slender working majority in Parliament?
Nobody knows the answer for sure.
Meanwhile, the Irish deputy prime minister said the Irish Republic would have “no choice” but to protect its place in the EU’s single market if the UK “forces a no-deal Brexit on everybody else”.
Also speaking on the BBC’s Andrew Marr Show, Simon Coveney warned that if the incoming Conservative prime minister chose to “tear up” the Brexit withdrawal deal, then “we’re in trouble”.
“That’s a little bit like saying, ‘Give me what I want or I’m going to burn the house down for everybody”.
Some 160,000 Conservative Party members are voting in a postal ballot to elect the next leader.
Ballots must be returned by 17:00 BST on Monday, with the winner of the contest due to be announced on Tuesday.