Thirty died in a double bombing on a mosque in Molai last night, Mohammed Kanar, from the National Emergency Management Agency (NEMA) said.
Three female bombers blew themselves up and killed four others in nearby Umarari early today, he added.
Locals, however, said the death toll was higher from both incidents and that more than 60 may have been killed in total.
Both areas are on the outskirts of the Borno state capital of Maiduguri, which has been increasingly targeted by coordinated bomb and suicide attacks in recent weeks.
The military was "well-positioned to meet the December deadline which they have been given", he said in a statement on Wednesday.
But with Maiduguri having been attacked four times this month alone, fresh questions will be asked about security in the city, where Boko Haram, which is allied to the Islamic State group in Syria and Iraq, was founded in 2002.
One area of the city, Ajilari Cross, has been hit three times in a month, killing more than 120. There have also been suicide attacks near the capital, Abuja.
Buhari said on Wednesday that Boko Haram's "ability to attack, seize, ravage and hold any Nigerian territory will have been completely obliterated" by December.
This week, he met the head of the US Africa Command, General David Rodriguez, as Washington announced the deployment of up to 300 military personnel to northern Cameroon.
The US military will provide intelligence, surveillance and reconnaissance expertise, the White House said, after similar multiple suicide attacks in Cameroon, Chad and Niger.
Guerrilla-style tactics against "soft" civilian targets such as mosques, markets and bus stations have increased.
Today, Nigeria's most senior army officer, Chief of Army Staff Lieutenant General Tukur Buratai, told troops "the next few days will be crucial" in the counter-insurgency.
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