"His replacement will be announced shortly".
Al Jazeera's Laurence Lee, reporting from London, said Johnson's departure left the British government in crisis mode.
Speaking to BBC radio yesterday, he said he hoped Britain would "resist very strongly any attempt to get any further concessions".
The departures raise the stakes for May, who believed that she had secured a hard-won agreement with her deeply divided cabinet of ministers on Friday to keep the closest possible trading ties with the EU.
Ms. May has faced a backlash over the plan from Brexit hardliners in her Conservative Party, who say it gives too many concessions to the European Union, but she has support from moderates and there has been no formal challenge to her leadership.
This last would likely be unacceptable to Brussels, thus killing May's plan, but Rees-Mogg told AFP his aim was only "to help the government stick to some of its earlier promises".
But he warned he felt "very strongly about the future of my country" and said Tories had "real concerns" about the direction of Brexit.
The UK will maintain a "common rulebook" for all goods traded with the European Union, including agricultural products.
Putin to attend World Cup final
Russia's run to the quarter-finals, which included a shock defeat of Spain in the last round, has been unexpected. Since the collapse of the Soviet Union, Russia had never managed to go past the group stage.
Some 64% of Britons do not trust her to run Brexit negotiations - up 31 percentage points from when we last asked the question in March 2017. May was forced to reshuffle her cabinet after Boris Johnson put in his papers on Monday.
There appears to be no immediate challenge to Ms May's leadership, as the Brexit hardliners simply do not have the numbers, and her Conservative Party seems set to weather this storm, despite deep divisions on the issue.
A long-awaited blueprint for the UK's relations with the European Union will be published later, with ministers vowing to deliver a "practical and principled" Brexit.
But Tory MPs lashed the Brexit blueprint for recreating the "worst parts of the EU" today as it was published for the first time.
May appointed staunchly pro-Brexit lawmaker Dominic Raab as the country's new Brexit secretary.Many pro-Brexit lawmakers were furious at what they saw as a sell-out of the clean Brexit they desire.
But May said talks with European Commission chief Jean-Claude Juncker, EU President Donald Tusk and other EU leaders at the weekend suggested her plan "can ensure that we move the negotiations on". Junior Minister Steve Baker, who worked under Davis, also resigned.
Under her proposal, a treaty would be signed committing the UK to "continued harmonisation" with EU rules - avoiding friction at the UK-EU border, including Northern Ireland.
Any leadership contest would also come from the Eurosceptic win of the Party, looking towards a Hard Brexit, whilst May and Raab negotiate for a deal with plenty of links to Europe.
This has angered some Conservative backbenchers, who say it will prevent the United Kingdom from having its own independent trade policy, and they want to see the Chequers blueprint rewritten.