Since weathering a bumbling leadership challenge from proponents of a clean split from Europe within her Conservative Party, Mrs. May has seen her popularity rebound. More people backed her staying on as prime minister than standing down in a YouGov poll this week, a reversal of the results from a week earlier.

But the draft deal itself remains deeply unpopular, with only 23 percent of people saying in a YouGov poll this week that they supported it, and only 3 percent saying they did so strongly.

Mrs. May ducked the thorniest questions put to her by radio callers on Friday, in a preview of what looked likely to be a week of public campaigning for the deal after European leaders vote on it in Brussels on Sunday.

She refused to discuss a backup plan if lawmakers in Parliament were to reject the draft deal in a vote expected by mid-December. She refused to say whether she would resign in that case. And she groped for a response to one of the most provocative questions conservative lawmakers have been wrestling with in recent days: Would the country be better off under Mrs. May’s plan or if it never left the union at all?

“What will make us better off is not so much about whether we’re in the European Union or not,” she finally said. “It’s about what we can do for our economy; it’s about what we can do for our prosperity.”