Bolton's comments came after meeting Prime Minister Boris Johnson at No 10 on Monday, reports the BBC.
The NSA said said the US supported a no-deal Brexit and added that Washington would propose an accelerated series of trade deals.
According to Bolton, deals could be done on a "sector-by-sector" basis, with an agreement on manufacturing being agreed first.
A bilateral agreement or "series of agreements" could be carved out "very quickly, very straight-forwardly", he said.
While saying that "both President Trump and I were leavers before there were leavers", he added that a trade deal for financial services and agriculture would not be the first to be agreed.
Bolton said "doing it in pieces" was not unprecedented and the US understood the importance and urgency of "doing as much as we can agree on as rapidly as possible because of the impending 31 October exit date".
When asked whether his proposed plan would follow World Trade Organisation rules, he said "our trade negotiators seem to think it is", the BBC reported.
He also criticised the European Union (EU) and accused them of treating voters like "peasants".
"The fashion in the EU when the people vote the wrong way from the way that the elites want to go is to make the peasants vote again and again until they get it right."
He made it clear that the US government "fully understands" that Brexit was the UK's first priority, and said issues like Iran, China, and the involvement of the Chinese telecoms firm Huawei in building the UK's 5G mobile infrastructure could be put off until after the UK leaves the EU.
( With inputs from IANS )