Information about the secret visit came from an Israeli who spoke on condition of anonymity, Haaretz reported.
The envoy delivered a message from British Prime Minister David Cameron that England believed Israel should be patient and allow Western sanctions against Iran to have maximum effect.
Britain believes there is still time to deal with Iran diplomatically, the source said.
The envoy met with a number of Israeli diplomatic and security officials.
The source said the envoy's visit, along with statements by top U.S. officials and a telephone conversation between Israeli Prime Minister Benjamin Netanyahu and German Chancellor Angela Merkel, have convinced the Israeli leadership to change their attitudes.
Netanyahu is now talking about setting "red lines" for Iran beyond which Israel and its allies would take action against Iran.
That idea has been repudiated by Washington. State Department spokesperson Victoria Nuland said recently that such talk "not helpful for the diplomacy."
The United States continues to push for more negotiations.
"We're watching very carefully about what they do, because it's always been more about their actions than their words," Secretary of State Hillary Clinton said during the weekend.