A Taliban spokesman on Twitter claimed the first attack, which killed 14 Nepali security guards working for the Canadian Embassy in Kabul in a massive blast that left their yellow minibus spattered with blood.
However Islamic State's affiliate in Afghanistan and Pakistan released a competing claim in which they named and pictured the alleged bomber, according to the SITE monitoring group, in what would be their most significant attack in the country.
The Taliban also claimed a second, smaller blast in south Kabul Monday that the interior ministry said killed one person.
The bombings were followed hours later by an attack on a market in the remote northeastern province of Badakhshan that authorities said killed at least 10 people, with the death toll set to rise.
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The wave of violence comes 10 days after Washington announced an expansion of the US military's authority to conduct air strikes against the Taliban, a significant boost for Afghan forces who have limited close air-support capacities.
Fourteen Nepali nationals were killed, the interior ministry said in a statement, with nine wounded -- five Nepali citizens and four Afghans.
The Canadian embassy in Afghanistan confirmed the "cowardly" attack in a tweet, and said that it had employed the guards.
The explosion could be heard across Kabul and a plume of smoke seen above the site of the blast on the Jalalabad road, a main route housing many foreign compounds and military facilities.
Nepal's prime minister K P Sharma Oli said his country "strongly condemns" the killings.
Taliban spokesman Zabihullah Mujahid claimed responsibility for the attack on social media, saying it was "against the forces of aggression" in Afghanistan.
The Islamic State claim, which SITE said was released on Twitter and Telegram, named the bomber as Irfanullah Ahmed and pictured him armed and dressed in combat fatigues in front of an IS flag.