"We consider such words from a Fox News correspondent unacceptable and offensive," Putin's spokesman Dmitry Peskov told journalists in a conference call.
"To be honest we would prefer to receive an apology addressed to the president from such a respected television company," the Kremlin spokesman added.
Fox News's Bill O'Reilly in an interview broadcast yesterday pressed Trump on Putin's alleged links to extrajudicial killings of journalists and dissidents, saying: "he's a killer though, Putin's a killer."
"Take a look at what we've done too. We've made a lot of mistakes," Trump added.
Asked to react to Trump's response, Peskov said "in this case I would prefer to leave this without comment."
Trump has prompted a political firestorm with his remarks on Putin in the interview broadcast ahead of the Super Bowl.
"I do respect him. Well, I respect a lot of people, but that doesn't mean I'll get along with them," Trump said.
Senate leader Mitch McConnell, speaking on CNN, said of Putin, "He is a former KGB agent (and) a thug."
Trump and Putin had their first call since the US president took office on January 28, which the White House called "a significant start to improving the relationship between the United States and Russia.
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