Johnson is expected to take office on Wednesday, vowing to deliver Brexit on October 31 come what may, in the face of intense opposition in parliament.
A resignation by Hammond, who has become an increasingly fierce critic of Johnson's approach, would show the depth of opposition Johnson may find himself up against as PM, observers said.
Former London mayor Johnson is the runaway favourite to win the governing Conservative Party's leadership contest on Tuesday and then replace Prime Minister Theresa May when she quits the premiership on Wednesday.
Observers say Hammond would never have expected to remain as Chancellor of the Exchequer under Johnson anyway.
But the fact that the second most senior figure in the government is pre-empting his removal in any cabinet reshuffle by the incoming prime minister is seen as significant.
"I'm sure I'm not going to be sacked because I'm going to resign before we get to that point," Hammond told BBC television.
"I understand that his conditions for serving in his government would include accepting a no-deal exit on the 31st of October.
"That is not something that I could ever sign up to."
The postal ballot of 160,000 grassroots party members is expected to return Johnson, 55, as the new Conservative leader over his contender, Foreign Secretary Jeremy Hunt.
Any remaining ballots must be delivered by Monday at 5:00pm (1600 GMT). Bookmakers give Hunt around a one-in-15 chance of victory.
- Threat to majority -
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The centre-right Conservatives command a razor-thin majority in parliament's lower House of Commons and Johnson's opponents -- both within and outside the party -- are keen to scupper his leadership.
Johnson has vowed to take Britain out of the European Union on October 31, with or without a divorce deal.
Opponents of Brexit, and especially of a no-deal departure, are plotting moves against Johnson.
Some Conservatives, Hammond included, have hinted they are prepared to bring down their own government rather than accept leaving the EU without an agreement.
"I cannot accept the idea of leaving with no deal on October 31," Hammond said.
Justice Secretary David Gauke also said Sunday he would quit the government if Johnson became prime minister.
The Sunday Times newspaper reported that up to six europhile Conservative MPs were considering defecting to the centrist, pro-EU Liberal Democrats should Johnson win -- leaving him without a Commons majority.
- 'Right side of history' -
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Pro-EU protesters rallied in central London on Saturday in anticipation of Johnson taking office, in a "No to Boris, yes to Europe" demonstration.
"I want to be on the right side of history," demonstrator Tamara Bishop told AFP.
Claire Payton said of Johnson: "He's going to be absolutely appalling."
US President Donald Trump said Friday that he got along well with Johnson, predicting he would fix the Brexit "disaster".
"He is going to do a great job," Trump told reporters.
After a month-long contest, the name of the new Conservative Party leader will be announced on Tuesday.
May will answer questions in parliament as prime minister for the final time at midday (1100 GMT) on Wednesday before heading to Buckingham Palace to tender her resignation to Queen Elizabeth II, the head of state.
The sovereign will then invite the new Conservative leader to form an administration.
Hammond will resign between prime minister's questions in parliament and May going to see the sovereign.
May, who took office following the seismic 2016 Brexit referendum, has vowed to be "absolutely" loyal to her successor.
The 62-year-old, who will remain in parliament as an MP, told the Daily Express newspaper that initially she would "take some time off, have a holiday and adjust to the new world".