Johnson's office said the prime minister and U.S. President Donald Trump's adviser "spoke about Brexit and a range of other issues - including Iran, Hong Kong and 5G" network security.
"We just ask that, as issues come up, we resolve them individually and we reserve the time to have a larger conversation on some of these important issues at a moment that is really right for the new government".
Trump had wanted to work with the May government on a trade deal but her government "didn't want to do it".
Bolton said it was an honor for him to meet with Johnson. "From the meetings we have had it is clear that they are not".
A senior government source said Downing Street expects the first showdown to happen on 9 September because the government is bound by law to provide a report on Northern Ireland on 4 September and to then debate it in parliament within five days.
Johnson wants the European Union to renegotiate the terms of Britain's exit ahead of an October 31 departure date, but the European Union says it will not alter the part of the deal Johnson says must be changed.
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But Bolton hopes to find a more friendly audience on the topic from the Johnson government.
The two leaders are to meet later this month at the G7 summit in Biarritz, France.
Boris Johnson's senior allies now believe the only person who can put a stop to Brexit is the Prime Minister himself.
The EU is refusing to re-open negotiations on the deal the bloc's 27 leaders signed past year with May. Since day one we have been pressing Government for clarity as regards future border crossing procedures and we have conveyed our concerns about the future to Monsieur Puissesseau on many occasions.
Johnson is seeking a deal with the European Union but has not ruled out suspending parliament to prevent lawmakers' attempts to block a no-deal exit. We are also making very detailed preparations at home to ensure the United Kingdom is ready to leave the European Union in all circumstances on 31 October.
He made it clear that the United States government "fully understands" that Brexit is 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.