Pete Buttigieg has criticised Hillary Clinton over controversial remarks about Tulsi Gabbard, one of his competitors for the Democratic presidential nomination, saying the former nominee should have provided evidence for her claim Gabbard is “a Russian asset”.

Clinton made the remarks about the military veteran and Hawaii congresswoman to former Obama aide David Plouffe’s Campaign HQ podcast.

Clinton suggested Gabbard was being “groomed” by Moscow to act as a spoiler in the 2020 race. US intelligence agencies agree Russia is seeking to influence the next election as it did Clinton’s losing battle with Donald Trump.

The Russians, Clinton said, have “got their eye on somebody who’s currently in the Democratic primary and are grooming her to be the third-party candidate”.

She added: “They have a bunch of sites and bots and other ways of supporting her so far.”

Clinton actually made the “Russian asset” remark about Jill Stein, the Green party candidate last time out, who rejected the remark this weekend. But an aide later confirmed the thrust of Clinton’s remarks referred to Gabbard, saying: “If the nesting doll fits.

“This is not some outlandish claim,” Nick Merill added. “This is reality.”

Buttigieg, the mayor of South Bend, Indiana, was asked about the controversy on the Sunday talkshows. He generally sidestepped but on CNN’s State of the Union he said: “Statements like that ought to be backed by evidence.

“I don’t know what the basis is for that. I consider [Gabbard] a competitor. I respect her service … I would prefer we had arguments in terms of policy.”

Gabbard shot back at Clinton on Twitter, calling the former first lady, senator and secretary of state the “queen of warmongers [and the] embodiment of corruption”.

On Saturday she went further, telling an NBC reporter as she campaigned in Iowa Clinton had started a “smear campaign that has been waged against me and my candidacy and my campaign from the very first day that we began.

“This smear campaign is coming from people like Hillary Clinton and her proxies, the foreign policy establishment, the military industrial complex, who obviously feel threatened by my message and by my campaign because they know that they can’t control me.”

Gabbard has qualified for all the debates but remains stuck in the low single digits in polling, way behind frontrunners Elizabeth Warren, Joe Biden and Bernie Sanders. Speaking with voters in Iowa, Gabbard was pressed about Clinton’s comments.

“What is this horrible thing that Hillary said about you?” one person asked.

Clinton’s comments were “divisive and despicable”, said Patricia McIntosh, 83, who told the Associated Press she liked Gabbard’s “anti-regime-change message”.

“I have no respect for Hillary Clinton at all,” McIntosh said.

Gabbard, who has faced criticism for visiting Syria’s Russia-backed president, Bashar al-Assad, said Clinton’s comments revealed “the truth that I have been experiencing for a long time now – which is that, because I have been trying to bring about an end to our country’s long-held foreign policy of waging one regime-change war after the next, I am labeled as a traitor.”

That message, she said, was sent to “every single American who speaks out for peace”.

Gabbard draws support from an eclectic base that loosely crosses over with that of Sanders. She has not broken 3% in a major poll and has claimed the Democratic National Committee (DNC) and corporate media are “rigging” the primary.

Gabbard threatened to boycott last week’s debate in Ohio, describing qualifying criteria as “arbitrary” and accusing party leaders “of trying to hijack the entire election process”.

Her unconventional policy positions – she supports drug decriminalization, for instance – have attracted support from unusual quarters. The congresswoman has for example found support from the former Trump campaign chief Steve Bannon and former Ku Klux Klan leader David Duke.

Another 2020 rival, the California senator Kamala Harris, has described Gabbard’s decision to meet with Assad as “un-American”.

Gabbard has assailed Harris over her record as prosecutor before entering elected office, in her words for putting “over 1,500 people in jail for marijuana violations and then laugh[ing] about it when she was asked if she ever smoked marijuana”.

Gabbard’s campaign has been promoted by Russian state-owned media and defended on Twitter by the Russian embassy – support she has refused to disavow.

During a stop in Iowa this weekend, one man offered her ushanka-style hat.

“It’s a Russian hat!” Gabbard said with a laugh, before taking a selfie.