Trump said he will bring up the topic at a meeting between the two leaders next week in Helsinki.
"His behavior this morning is another profoundly disturbing signal that the president is more loyal to President Putin than to our North Atlantic Treaty Organisation allies", Pelosi and Schumer said in a joint statement.
"The indictment charges twelve Russian military officers by name for conspiring to interfere with the 2016 presidential election", Rosenstein announced at a press conference at the Department of Justice just after 12:00 PM eastern.
"Former Secretary of State John Kerry issued a scathing statement about what he called a "disgraceful", "disgusting" and downright "strange" appearance by President Trump at the North Atlantic Treaty Organisation summit in Brussels".
Speaking of topics on agenda, the Russian official said that it remains unknown what exact issues will be touched upon at the top-level meeting, adding that Moscow is ready to "consider literally all key issues on the bilateral and worldwide agenda".
Kremlin and White House officials maintain there's much to be gained on both sides from the summit.
Pompeo - formerly director of the Central Intelligence Agency - added that it's "very important" that Trump and Putin meet.
Donald Trump's portrayal of North Atlantic Treaty Organisation as an alliance in crisis has raised concern that the USA president's nagging criticism might erode US public support and risk America's commitment to collective defense, diplomats said.
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The indicted group has been charged hacking the Democratic National Committee and Hillary Clinton's presidential campaign during the 2016 presidential election along with leaking stolen information to the public.
Just days earlier in Brussels, Trump offered startlingly cutting criticism of one of the closest U.S. allies - accusing Germany of being "a captive of Russia" - before insisting hours later that he has "a very, very good relationship with (German Chancellor Angela Merkel)" and a "tremendous relationship with Germany".
"I don't think anything gets the message across to Trump, but I hope demonstrations like these encourage people in the States to fight the Trump regime", said Eli Roth, a 56-year-old from California who is living in Edinburgh.
But Trump has also repeatedly said he takes Putin at his word when he denies any involvement. "But I will absolutely, firmly ask the question". And John McCain, a leading Republican senator and Trump critic, said Trump must hold Putin accountable or not proceed with the meeting. "This isn't good for the United States and there are people across the aisle - as the Senate vote yesterday clearly showed - who know it and need to say it", Kerry said on Twitter. Pure stupidity. But it makes it very hard to do something with Russian Federation.
"The concern is that this is not just noise to disregard, because perception is reality and this president may be with us for another six years", the senior source said.
Scuffles broke out at one point as attendees - who chanted slogans including "We love you Donald" - passed a nearby anti-fascism protest.
The indictments are the most detailed allegations so far of how units of Russia's GRU tried to influence the 2016 election by stealing Democratic emails, then releasing them in ways meant to dominate news headlines as voters made up their minds.