Godane, who once appeared in an al-Shabaab video pledging his allegiance to Osama bin Laden, gave his backing to last September's Westgate atrocity, in which 67 people were killed when gunmen stormed a shopping centre in Nairobi.
In a statement released shortly after the Westgate attack, he threatened further terrorism in Kenya unless it agreed to pull its troops out of Somalia, where they have been trying to oust al-Shabaab from its southern heartlands.
The attack was one of the worst terrorist acts in Kenyan history. Godane, who also went by the name Abu Zubayr, was listed by the US State Department as one of the world's most-wanted terrorist fugitives, with a US$7 million ($8.4 million) bounty on his head.
Somali officials said the first they knew about the drone attack was when government and African Union forces patrolling in the area heard a series of explosions. The strike came as local al-Shabaab forces were on the back foot from a government offensive aimed at seizing key ports.
Godane is known to have taken security precautions after the killing of al-Shabaab leader Adan Hashi Ayro in a US missile strike in 2008.
Godane's death would be a severe blow for the group, which has lost much territory in the past two years after offensives by Somalia's US- and UN-backed Government.