Hunter Biden would also not work for any foreign-owned companies if his father were elected, his lawyer said.
Mr Trump has called on China and Ukraine to investigate Joe Biden over his son's activities in both countries, without any evidence of any wrongdoing.
The issue is central to an impeachment inquiry against the US president.
Mr Trump and his supporters have accused Joe Biden of abusing his power when he was vice-president to pressure Ukraine to back away from a criminal investigation that could have implicated his son who then worked for a Ukrainian energy company.
He has also accused Hunter, 49, of using his position of influence to get China to invest in a fund he was involved in.
The impeachment inquiry begun by opposition Democrats in the US House of Representatives centres on allegations that the president broke the law by soliciting help from a foreign power - Ukraine - for electoral gains.
What did Hunter say?
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