Johnson will offer the three leaders the option to either help him deliver a new deal this week, or to agree on a friendly version of a no-deal Brexit by Oct. 31, the newspaper said, citing a source familiar with the conversations.
"He'll be talking to Merkel, Macron and Juncker by the end of Monday to see if there's agreement on a 'landing zone' for Northern Ireland and customs," the source was quoted as telling the newspaper.
"The alternative is to agree a friendly version of no deal and finish it that way."
Security chiefs have convinced Johnson that a no-deal Brexit will heighten the danger of extremist attacks in Northern Ireland and on the mainland, along with sectarian violence in cities such as Glasgow, according to the report.
As a result, the British prime minister desperately wants a Brexit deal, the Sunday Times reported.
"Any one of these risks we could cope with, but taken collectively they would be a massive challenge to the UK state and no one would choose to go down that route," Johnson told a senior Conservative in a private conversation, according to the newspaper.
A European Union source cited by the Sunday Times said the chances of a deal at Thursday's EU summit in Brussels were "50-50".
On Friday, Johnson had declined to say whether Northern Ireland will stay in the EU customs union after Brexit.
A Downing Street source told Reuters on Saturday that Britain remains a long way from agreeing a final Brexit deal and the next few days will be critical if it is to agree departure terms with the EU.