Journalist Ben Jacobs tweeted that Mr Gianforte had "body slammed" him, breaking his glasses.
The front runner's campaign said that Mr Jacobs had initiated the incident by grabbing Mr Gianforte's wrist.
But other journalists said they saw the reporter show no physical aggression.
The incident took place on Wednesday, the eve of a special election to fill a vacant seat in the House of Representatives.
The Guardian released an audio recording of the clash, in which crashing sounds can be heard when Mr Jacobs presses the candidate for a response to a question.
Mr Gianforte then shouts "I'm sick and tired of you guys" and tells the reporter to "get the hell out of here".
Alicia Acuna of Fox News was in the room with two colleagues preparing for a television interview with Mr Gianforte at the time.
She said Mr Jacobs entered the room, put a voice recorder up to Mr Gianforte's face and asked about the Congressional Budget Office report on the American Health Care Act.
When Mr Jacobs persisted, "Gianforte grabbed Jacobs by the neck with both hands and slammed him into the ground behind him", she wrote.
"Faith, Keith and I watched in disbelief as Gianforte then began punching the reporter. As Gianforte moved on top of Jacobs, he began yelling something to the effect of, "I'm sick and tired of this!", she added.
"At no point did any of us who witnessed this assault see Jacobs show any form of physical aggression toward Gianforte," she said.
Mr Jacobs later told MSNBC he then went to a hospital to have his elbow X-rayed, because he landed on it during the incident.
He said he believed the campaign staff were unhappy about previous dealings with Guardian colleagues.
Three of the state's largest newspapers, the Billings Gazette, the Helena Independent Record, and the Missoulian, have since their endorsements of Mr Gianforte.
"We are also sick and tired - of Gianforte's incessant attacks on the free press," said the Independent Record.
The Missoulian noted that as the eyewitness report came from right-wing Fox News, the incident could not be construed as liberal "fake news".
Mr Gianforte's campaign spokesman said Mr Jacobs entered the office without permission and "aggressively shoved a recorder in Greg's face and began asking badgering questions".
The candidate, "attempted to grab the phone", then "Jacobs grabbed Greg's wrist and spun away from Greg, pushing them both to the ground", Shane Scanlon wrote in a statement.
"It's unfortunate that this aggressive behaviour from a liberal journalist created this scene at our campaign volunteer BBQ," he added.
Gallatin County Sheriff's Office said that after an investigation a citation was issued to Greg Gianforte on Wednesday night for misdemeanour assault.
This is a lesser offence than felony assault, for which "the nature of the injuries did not meet the statutory elements", Sheriff Brian Gootkin said.
The maximum penalty for the offence is a $500 (Â£385) fine and a six-month jail term.
Records show that Sheriff Gootkin has donated $250 to Mr Gianforte's congressional campaign.
The vacancy in Montana arose after President Donald Trump appointed the previous holder of the seat, Ryan Zinke, as interior secretary.
Democrat Rob Quist, a banjo-playing folk singer and first-time candidate, is battling Mr Gianforte, a multimillionaire technology executive and Christian conservative.
In Washington, California Republican congressman Duncan Hunter said it was not appropriate behaviour.
"Unless the reporter deserved it", he reportedly added.
Speaker of the House Paul Ryan, the highest ranking Republican in Congress, said Mr Gianforte should apologise to the reporter.
The Republican candidate in a special congressional election in Montana, Greg Gianforte, has been charged with a misdemeanour for assaulting a Guardian reporter.