Priti Patel, the worldwide development secretary, also said that the meeting and others with senior Israeli officials had been conducted without the knowledge of the British Foreign Office and apologised for failing to follow "usual procedures".
Meanwhile, the Times has reported that Patel has been accused of breaching the ministerial code by letting a leading figure in a corporate lobbying group, Lord Polak, sit in on her secret meetings with members of the Israeli government.
In her resignation, Ms Patel apologised and said her actions "fell below the standards of transparency and openness that I have promoted and advocated".
In reply, May said in a letter to her: "Now that further details have come to light, it is right that you have made a decision to resign".
"I offer a fulsome apology to you and to the Government for what has happened and offer my resignation", Patel said in a letter.
"But what I then found out Priti Patel mentioned that meeting and that Number 10 told her not to put that meeting on the list because it would embarrass the Foreign Office".
Israeli Public Security Minister Gilad Erdan tweeted about a meeting he had with Patel in the British parliament on September 7, describing it as "wonderful".
On 18 September, while in New York, Patel met Yuval Rotem, an official from the Israeli foreign ministry.More news: WWE SmackDown: SHOCK Title Change In UK
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On returning from Israel, Ms Patel commissioned work by DfID on disability, humanitarian and development partnerships between Israel and the UK. "I am sorry for this and I apologize for it".
"Number 10 must answer questions about their complicity in this scandal".
Patel, who was born in London to a Ugandan Indian immigrant family, is - or was - a rising star on the right wing of the Conservative Party and a zealous advocate of leaving the E.U. Her political idol is Margaret Thatcher, who served as prime minister from 1979 to 1990 and died in 2013.
The statement added that the Foreign Office was aware of the visit while it was underway, but was not informed about it in advance.
Ms Patel was also reported to have visited the Golan Heights, territory disputed between Israel and Syria, which British officials tend to avoid.
Under British protocol, a cabinet minister would normally organise meetings through the foreign office and be accompanied by officials, and visits with Israelis would typically be balanced with meetings with Palestinians.
Labour has already demanded an investigation into Patel's meetings with the Israeli government, claiming they involved four "serious breaches" of the ministerial code.
The Sun newspaper reported yesterday that Patel had also failed to disclose that she had met the director general of Israel's Foreign Ministry, Yuval Rotem, in NY and Public Security Minister Gilad Erdan in London.