Harley sales already had slipped 8.7 percent in the first half of 2018 while its main American competitor, Polaris and its Indian brand, saw a 4 percent gain, according to industry analyst Gerrick L. Johnson at BMO Capital Markets. While that didn't cover the period from when the president's dispute with the company began, Johnson said recent evidence suggests the fight is adding to Harley's woes.

Harley-Davidson 's spat with President Donald Trump seems to be having a significant impact on the iconic motorcycle manufacturer, according to an analyst report. Trump backed a possible boycott of the Wisconsin-based motorcycle manufacturer on social media.

Harley riders have been trading in their "hogs" in increasing numbers as the company fights through a public relations nightmare created when it said it was shifting some production overseas, Johnson said. The Milwaukee-based manufacturer said it was doing so in response to retaliatory tariffs abroad due to Trump's duties on steel and aluminum imports from other countries.

"While President Trump's assessment of HOG's actions in his tweets were often factually inaccurate, the damage has been done," said the BMO report. "Dealers are feeling an impact, and we find that the impact has become more acute over time."

While both companies have tried to tamp down the president's influence on motorcycle sales, Johnson said the impact is unlikely to be a coincidence. Back in August, Trump used his Twitter account to lambaste the company for its production shift and seemed to back a boycott he said was already underway. It's not clear whether a boycott ever began.

Harley tweet

"While President Trump's assessment of HOG's actions in his tweets were often factually inaccurate, the damage has been done," Johnson said in a research note. "Dealers are feeling an impact, and we find that the impact has become more acute over time."

Johnson said he was cutting Harley's stock from outperform to market perform and knocked the price target to $45 from $52, which still implies 11.5 percent upside from Thursday's close. Harley shares were off by 2percent Friday morning after sliding 3.1 percent in Thursday's broad market sell-off. The stock has tumbled just over 20 percent year to date.

CNBC has reached out to Harley-Davidson for comment. The company reports earnings on Tuesday (an earlier version of this story said Monday).

It's not so much a problem with Harley's products, Johnson said. He added that the company has continued to bring good bikes to market, but to little avail.

"For the first time in 10 years of covering powersports we have seen a manufacturer develop innovative new products that are significantly better than what they replace, and yet not seen an increase in demand," he wrote.

But the company, according to the analysis, has failed to capture young riders, due in part to overpricing, and it managed the public relations poorly around the production move and Trump's subsequent reaction. Indian has had some success in attracting younger riders, BMO said.

"The overlap between Donald Trump supporters and Harley riders is significant," Johnson said. "Thus, it's disconcerting to have the President call on these consumers to boycott the brand. In response to new European tariffs on American motorcycles, HOG may have had the right financial intentions but the way it communicated its strategy was a public relations debacle."

Johnson suggested the company shouldn't have been as public as it was in announcing the move and shouldn't have decided to "poke" the president when doing so.

The issues this year continue a trend in which Polaris and Indian have been able to take market share. The company still only holds 7 percent of the market for "heavyweight" brands, compared with 50 percent for Harley-Davidson but says it has about a 20 percent share in the midsize market, which is geared toward younger riders.

In addition, Indian has grown year-over-year sales in 42 of 46 months, while Harley has done so just 12 times during the period.

Since the battle with Trump, Johnson said he has found "that a majority of dealers feel they have lost at least some sales."

"It should be disconcerting to HOG investors that most Indian dealers we speak with are seeing an uptick in Harley trade-ins for whatever the reason may be," the analyst wrote.