Mr Trump tweeted to thank Gen McMaster, saying he had done an “outstanding job & will always remain my friend”.
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.