The claim of the al-Qaeda-linked group on Friday could not be verified.
“While the mujahedeen have executed the serving KDF [Kenya Defence Force] soldier, there is still a chance of securing the release of the remaining five prisoners,” the al-Shabab said in a statement.
Last month the issued a February 14 deadline ordering Kenya – whose troops are fighting the Shebab inside Somalia – to release “all Muslim prisoners held on so-called terrorism charges in Kenya”.
Al-Shabab, who have previously released videos of Kenyan civil servants they have kidnapped, have said they would execute five hostages within three days unless the Kenyan government buckles to their demands.
Kenya has been hit by a spate of attacks including hand grenade and bombs since it invaded southern Somalia in late 2011 to attack Shebab bases, following a string of kidnappings inside Kenya blamed on the armed group.
Many of the attacks in Kenya – including hand grenade blasts in the capital Nairobi – are blamed on Shebab supporters or Kenyan sympathisers, although the Shebab have not claimed the attacks themselves.
But the once powerful al-Shabab are on the back foot inside Somalia, having fled a string of key towns in the face of a 17,000-strong African Union force – which includes Kenyan troops – which is also fighting alongside Somali soldiers.
Ethiopian troops are also battling al-Shabab in the southwest of Somalia.
On Thursday, AU troops and government forces seized the towns of Janalle, Aw Dhigle and Barire, about 80 kilometres southwest of the capital Mogadishu, the latest Shebab bases to fall.
However, al-Shabab remain a potent threat, still controlling rural areas as well as carrying out guerrilla attacks in areas apparently under government control.