The Kremlin aide, Yuri Ushakov, said Trump had suggested on the call that the summit could be held at the White House. He is scheduled to attend the summit meeting of the North Atlantic Treaty Organization July 11 and 12 and to visit Britain on July 13.
Such a summit is likely to irritate USA allies who want to isolate Mr Putin, such as Britain, or who are concerned about President Trump's attitude towards Russian Federation.
Putin and Trump had two brief meetings on the sidelines of worldwide summits previous year and both men spoke of a mutual desire for a full-fledged one-on-one summit during a March telephone call.
Bolton said that he would not address previous statements he had made and reiterated that Trump will "raise the full range of issues" between the United States and Russian Federation when the presidents meet in July. Bolton is expected in Moscow on Wednesday for talks with Russian Foreign Minister Sergey Lavrov and possibly Vladimir Putin.
"Even in earlier days, when our countries had differences, our leaders and their advisors met and I think that was good for both countries, good for stability in the world, and president Trump feels very strongly on that subject", replied Bolton.
Mr Trump told reporters he will discuss the war in Syria and the situation in Ukraine with his Russian counterpart.
Asked whether Russian Federation should be part of the G7 group of most industrialized nations, Pompeo said Trump should speak for himself, adding, "The president deeply believes that having Russian Federation be part of these important geostrategic conversations is inevitable, and there's a long history of that". "So maybe something positive will come out of it", Trump said.
"To say there is a single issue that has caused there to not to be a warm relationship between the two countries is a misnomer", Secretary of State Mike Pompeo told CNN's Elise Labott on Sunday. Anor option considered in Moscow is Vienna.
Trump has denied any collusion with Russians during his 2016 campaign, and Bolton echoed that denial.
Interested in Donald Trump? The lack of American reaction to Syrian military offensive to south with help of Russian aviation makes Russian analysts think that Washington is ready for concessions. However, an array of new Russian sanctions has been imposed during his presidency.