The US has imposed tariffs on steel and aluminum against the European Union and other allies, sparking retaliation.
"I think the president can put this on the table and say, 'This is a serious matter that we need to talk about,"' said Bolton, adding that asking for the indicted Russians to be turned over would have the opposite effect.
Mr Trump also says he "hadn't thought" about asking Mr Putin to extradite the dozen Russian military intelligence officers indicted this past week in Washington on charges related to the hacking of Democratic targets in the 2016 U.S. presidential election, but says "certainly I'll be asking about it".
"Russia is a foe in certain respects", he said. The US leader added that he doesn't necessarily view them negatively. But that doesn't mean they are bad. It doesn't mean anything.
Trump, in Orwellian spin, brands European Union a foe.
But after being given the idea by his interviewer, Trump said, "Certainly I'll be asking about it".
The visit has been controversial because of a newspaper interview in which he said the USA would probably not give the United Kingdom a trade deal under the terms of Prime Minister May's Brexit plans, and also said Europe was "losing its character" because of immigration from Africa and the Middle East.
Echoing comments made ahead of a North Atlantic Treaty Organisation summit in Brussels, Trump then turned his attention to Germany, criticizing their defense spending contributions as well as a deal to build a natural gas pipeline to Russian Federation. "I think it's ridiculous", he told CBS.
"I also think it is very bad thing for Germany".
He also accused Germany of "waving a white flag".
A probe over allegations of Russian meddling in the 2016 USA presidential election has clouded Trump's presidency.
From aboard Air Force One, Trump complained in tweets that he wasn't getting enough credit for his meeting with Kim and railed that "Much of our news media is indeed the enemy of the people" as he headed to sit down with Putin. Look at the millions of cars they sell in the US.
There have been calls in the US for Mr Trump to cancel his meeting with Mr Putin over the indictments, announced on Friday by US Deputy Attorney General Rod Rosenstein. And, you know, I love those countries. There was widespread anger in Germany in 2013 when it was revealed the US National Security Agency tapped Merkel's phone.
US President Donald Trump has described the European Union as a foe on trade.
"Despite what you have heard from the fake news, I had a great meeting with German Chancellor Angela Merkel".
German Foreign Minister Heiko Maas has warned Trump against making any unilateral deals with Russian Federation that come with a cost for the United States' Western allies. "Big mistake made all over Europe in allowing millions of people in who have so strongly and violently changed their culture!"