"You can only ask so many times".
The White House said the two leaders will discuss US-Russian relations and "a range of national security issues".
Trump is due to attend the July 11 and 12 North Atlantic Treaty Organisation summit in Brussels before heading to Britain to meet with Prime Minister Theresa May and Queen Elizabeth II on July 13. The summit's date will give Putin a chance to attend the July 15 closing ceremony of the soccer World Cup which his country is hosting.
The two leaders have met twice before on the sidelines of worldwide gatherings and spoken at least eight times by phone.
Bolton himself suggested Wednesday that the administration had no concrete goals for the meeting in Helsinki, beyond building trust and understanding between the two leaders.
The meeting comes amid FBI Special Counsel Robert Mueller's probe into alleged Russian interference in the 2016 election. Trump denies wrongdoing and calls the investigation a "witch hunt".
US-Russian relations have hit lows not seen since the Cold War over Moscow's annexation of Crimea, its involvement in eastern Ukraine and allegations of political meddling, as well as disagreement over the Syrian conflict.
US President Donald Trump said on Wednesday he could meet with his Russian counterpart Vladimir Putin during his trip to Europe. "Do you hope he says to Vladimir Putin, "I don't believe you when you say Russian Federation did not meddle in the U.S. Election"?"
With allegations still swirling about Russian "meddling" in the 2016 election, President Trump has been very cautious with his diplomatic ties with Russian officials.
"A lot of people have said or implied over time that a meeting between President Trump and President Putin would somehow prove some nexus between the Trump campaign and the Kremlin, which is complete nonsense", Bolton said.