Germans should "live together with the animals, in the way you are trying to tell us to," Mokgweetsi Masisi told German daily Bild.
"This is not a joke," said Masisi, whose country has seen its elephant population grow to some 130,000.
Botswana has already offered 8,000 elephants to Angola and another 500 to Mozambique, as it seeks to tackle what Masisi described as "overpopulation."
"We would like to offer such a gift to Germany," Masisi said, adding that he would "not take no for an answer."
The African leader argued that conservation efforts have led to an explosion in the number of elephants and that hunting is an important means to keep them in check.
The herds of elephants were causing damage to property, eating crops and trampling residents, he told the German paper.
Germany's environment ministry earlier this year raised the possibility of stricter limits on the import of hunting trophies over poaching concerns.
But a ban on the import of hunting trophies would only impoverish Botswanans, Masisi said.
Botswana banned trophy hunting in 2014 but lifted the restrictions in 2019 under pressure from local communities.
The country now issues annual hunting quotas.
A spokeswoman for the environment ministry in Berlin said Botswana had not raised any concerns with Germany on the matter.
The ministry however remained in talks with African countries affected by import rules, including Botswana, the spokeswoman said.
"In light of the alarming loss of biological diversity, we have a special responsibility to do everything to ensure the import of hunting trophies is sustainable and legal," she said.
Germany was one of the largest importers of hunting trophies in the European Union, she said.
African elephant hunting trophies already require import authorisation under current rules, she added.
Discussions within the EU on harsher import restrictions are focused on extending the list of protected species, she said.