Later he was to meet at No. 10 Downing Street with Edward Lister, who is Johnson's chief strategic adviser.
A central message Bolton was making is that the United States will help cushion Britain's exit from the European Union with a free trade deal that is being negotiated by US Trade Representative Robert Lighthizer and his British counterpart, Liz Truss.
Bolton said he had received reassurances from Johnson that Britain was "concerned about not having any compromise in the security of telecommunications in the 5G space".
Trump had wanted to work with the May government on a trade deal but her government "didn't want do it".
As President Donald Trump's administration predicts the United States and the UK will be more in sync under Johnson - who has been referred to as the UK's Trump - than the countries were under former Prime Minister Theresa May, Bolton will explicitly articulate Trump administration policies and identify areas where the nations can work together, particularly when it comes to being tough on Iran and Huawei, a senior administration official explained to reporters traveling with Bolton.
Trump expressed appreciation for Britain's "steadfast partnership" in tackling global challenges in a call with Johnson, a White House statement said on Monday.
But a clean break with the European Union would allow the UK to immediately launch negotiations on a free trade agreement with the United States. Later in the day he planned to meet Sajid Javid, the new chancellor of the exchequer.
Britain has been under pressure to take a more hawkish approach to Iran since its seizure of an Iranian oil tanker near Gibraltar in early July - it was suspected the ship was marshaling oil to Syria.
Trump has also pushed Britain to get tougher on China's Huawei out of concern its next-generation 5G technology represents a national security risk.
But the British government has insisted no decision has been made about the use of Huawei's equipment in the United Kingdom.
Mr Bolton also referenced Mr Johnson's willingness to participate in Operation Sentinel, which aims to beef up the military presence in the Gulf in the face of tensions between the West and Iran, saying he was "pleased" as this "reflects a change from the prior government".
Britain's decision on Huawei - a private firm that Washington claims is obliged to spy for the Chinese government - has been repeatedly delayed due to mixed signals from Trump about his own administration's next steps.