Mr Bolton, who has backed attacking North Korea and Iran, told Fox News his job would be to ensure the president has "the full range of options".
He becomes Mr Trump's third national security chief in 14 months.
Gen McMaster is the latest high-profile departure from the White House.
Last week, Mr Trump fired Secretary of State Rex Tillerson by a tweet, replacing him with former CIA Director Mike Pompeo.
Mr Bolton's appointment does not require US Senate confirmation. He will take the job on 9 April.
Responding to the move, Mr Bolton said he was looking forward to working with President Trump and his team "to make our country safer at home and stronger abroad".
Who is John Bolton?
Mr Bolton, 69, has been a foreign policy hawk in Republican circles for decades, having served in the administrations of Ronald Reagan, George HW Bush and George W Bush.
The second Bush appointed him as US envoy to the UN, during which time diplomats privately criticised Mr Bolton's style as abrasive.
A strident neo-conservative, Mr Bolton helped build the case that Saddam Hussein possessed weapons of mass destruction, which turned out to be wrong.
Known for his walrus moustache, Mr Bolton does not appear to have greatly moderated his views since his last spell in government.
He stands by the invasion of Iraq and has advocated in newspaper op-eds using military force against North Korea and Iran.
More about: Trump
