The AU mission in Somalia (AMISOM) will be boosted by 6,235 troops, taking the total number to 23,966, the AU's Peace and Security Council said in a statement.
The request must be approved by the UN Security Council before fresh forces are deployed.
The decision comes in the wake of last month's deadly attack in a Nairobi shopping center which was claimed by the Shebab.
The AU said "renewed efforts" were needed to fight the Shebab in Somalia, and that it was deeply concerned by "the serious threat that Al Shebab continues to pose in Somalia and in the region."
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AMISOM troops have clocked successes in recent months, and have taken over major cities including the capital Mogadishu and the key port town Kismayo.
However, key Shebab strongholds remain, including rural southern and central Somalia, while another faction has dug into remote and rugged mountains in the northern, semi-autonomous Puntland region.
AU forces are battling alongside Ethiopian troops who invaded the country in 2011 to support Somali government troops to fight Shebab.
Somalia has been ravaged by almost uninterrupted conflict since 1991.