House Speaker John A. Boehner told reporters Thursday he continued to seek bipartisan consensus for an immigration overhaul, but demurred on the specifics.

For any legislation to pass muster, the Ohio Republican reiterated, it would have to contain stronger border security enforcement mechanisms. How much stronger? That’s a question Boehner isn’t yet prepared to answer, but he knows the Senate bill isn’t strong enough.

“This week, the Congressional Budget Office agreed,” Boehner continued. “It found that illegal immigration would only drop by 25 percent under the Senate plan, and that will not cut it.”

He declined to comment on the latest Senate immigration compromise on the issue — the Corker-Hoeven amendment — saying he hadn’t studied it.

Boehner did acknowledge that the rest of the CBO report, which found that the Senate bill would result in cost-savings and deficit reductions, was worth taking a closer look at — and he said he would delegate that task to House Budget Chairman Paul D. Ryan of Wisconsin.

“I haven’t studied the report. As matter of fact, I asked Chairman Paul Ryan this morning to do his own analysis on what they’ve done,” Boehner said at Thursday’s briefing. “I want to get to the bottom of it because, if in fact these numbers are anywhere close to being accurate, it would be a real boon for the country.”

Boehner also would not answer the question of whether he would — or whether he thought his conference would — support an immigration overhaul bill that also contained a pathway to citizenship for the country 11 million undocumented immigrants. Many Republicans have said they don’t want such a pathway to be included in legislation until there was certainty that the borders would be properly secured.