“We tell you that no Dane will foul the land of Sudan again,” he told thousands at a rally organised by his ruling party on Wednesday.
“We are capable of delivering the decisive response … boycotting personalities and companies,” he said.
Al-Bashir did not say if the decision included diplomats. Danish consulate officials in Khartoum were not available for comment.
Last Friday, several hundred Sudanese demonstrated in the capital to urge that diplomatic relations with Denmark be cut and for its products to be boycotted.
The latest protests were sparked when Danish newspapers reprinted a drawing featuring the Prophet Muhammad wearing a bomb in his turban.
The caricature was one of several first published in early 2006 which sparked violent protests in many Muslim countries.
Those protests culminated with the deaths of dozens of people in Nigeria, as well as the torching of Danish diplomatic offices in the Syrian capital Damascus and the Lebanese capital Beirut.
The Danish newspapers republished the drawings a day after police in Denmark foiled an alleged plot to murder the cartoonist.
The newspapers said that they republished the cartoons in solidarity with the cartoonist and in defence of freedom of expression.
Islam forbids any physical representation of the Prophet Muhammad as idolatrous.