Mr Bashir is wanted by the International Criminal Court (ICC) for alleged war crimes in Darfur.
Sudan has ordered the Kenyan ambassador to leave the country within 72 hours.
It has also ordered the Sudanese ambassador in Kenya to return to Khartoum.
Mr Bashir was the first head of state to be indicted by the ICC, which accused him of genocide and crimes against humanity in the Sudanese region of Darfur.
He denies the charges, saying they are politically motivated.
Chinese support
The High Court in Nairobi on Monday issued the arrest warrant for President Bashir after Kenya allowed him to visit the country in August in defiance of an ICC warrant for his arrest.
Kenya's Foreign Minister Moses Wetangula told the BBC he expected Attorney General Githu Muigai to appeal against the ruling.
Mr Bashir enjoyed "sovereign immunity" and the African Union (AU) had decided that member states should "disregard" the ICC's arrest warrant, he said.
"So it's not just about Kenya and Sudan and our diplomacy. It's also about our duty as an African country," Mr Wetangula told the BBC's Focus on Africa programme.
In his ruling, Judge Nicolas Ombija said Mr Bashir's arrest "should be effected by the attorney general and the minister for internal security should he ever set foot in Kenya".
The case was brought by a non-governmental organisation, the International Commission of Jurists (ICJ).
Kenya is a party to the treaty which established the ICC in 2002. But like most African countries, it has refused to enforce the ICC warrant for Mr Bashir's arrest.
The AU has lobbied for the arrest warrant to be deferred, accusing the ICC of only investigating alleged war crimes in Africa and arguing that arresting Sudan's president would hamper the search for peace in Darfur.
Malawi and Chad are among other African countries that Mr Bashir has visited in defiance of the arrest warrant.
The BBC's James Copnall in Khartoum says President Bashir's international reputation reached its lowest point after the ICC issued an arrest warrant against him. But he has received support from several Arab and African countries, and from China.
Some 2.7m people have fled their homes since the conflict began in Darfur in 2003, and the UN says about 300,000 have died - many from disease.
Sudan's government says the conflict has killed about 12,000 people and the number of dead has been exaggerated for political reasons.
* Kenyan Bantu at it again, This time Pissing off the sudanese.
Meanwhile Not long ago:
Kibaki and A bashir Cutting a cake in their happy days.lol
Lest we forget:
