Speaking during a press conference in the Republic of Georgia, Bolton said he offered a formal invitation to the Russian leader during his visit to Moscow this week.
Russian president Putin had last visited the White House in September 2005.
President Trump held a summit with Mr Putin in Helsinki, the Finnish capital, in July, and then issued Mr Putin an invitation to visit Washington in the autumn.
Trump's performance and comments at the summit were heavily criticized, including by members of the president's own party. "We have invited President Putin to Washington".
"We may - it's being discussed right now", the president told reporters at the White House.
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Trump's national diversity council in 2016, said the suspect's political views shouldn't be held against the president. Sayoc was being held at an Federal Bureau of Investigation processing centre in Miramar, Florida, CNN said.
Trump has said it is in USA interests to establish a solid working relationship with Putin.
Earlier this month, the Russian leader said "playing the Russian card" was a convenient instrument in U.S. politics ahead of midterm election in November.
The topic of Mr Putin visiting the U.S. is a highly charged one, because USA intelligence agencies allege that Russian Federation interfered in the 2016 presidential election to help Mr Trump win.
Speaking at a press conference in Moscow, Putin told reporters Russian Federation would respond in kind to the deployment of United States nuclear missiles to European countries. "I think we probably will".
The INF treaty, signed by ex-US president Ronald Reagan and Soviet leader Mikhail S. Gorbachev in 1987, prohibits the development and deployment of land-based intermediate-range cruise missiles. "In other words, there is no difference", said trump.", said trump.