The Lincoln Project, a Republican anti-Trump political action committee, released an ad accusing the president of knowing about the intelligence and doing nothing.
“How can Trump lead America when he can’t even defend it?” the organisation asks in the caption.
Another advert posted by FindAClearTruth on Sunday has over 21,000 retweets and 25,000 likes and focuses on the same accusation against the president.
The video opens with clips of Mr Trump insisting that he is aware of everything going on within the US military, followed by reports of the intelligence.
The footage then shows historical clips of Mr Trump stating he “knows nothing about Russia”.
The adverts come after intelligence gleaned from US military interrogations of captured militants in recent months suggested that Russian bounties offered to Taliban-linked militants to kill coalition forces in Afghanistan.
Controversy in Washington has emerged over the weekend around denials by Mr Trump and his aides that they had been briefed on the intelligence.
The bounties are believed to be linked to the deaths of several US service members, but it is unclear exactly how many Americans or coalition troops from other countries may have been targeted.
Both The New York Times and the Associated Press reported that US intelligence personnel told the president about Russia’s actions in Afghanistan some months ago.
The president has denied being briefed by US intelligence officials on the intelligence.
“Intel just reported to me that they did not find this info credible, and therefore did not report it to me or Mike Pence,” Mr Trump wrote.
The reports, he added, were “possibly another fabricated Russia Hoax, maybe by the Fake News New York Times Books, wanting to make Republicans look bad!!!”
Democratic presidential nominee Joe Biden condemned Mr Trump on Saturday and said that “his entire presidency has been a gift to Putin”.
“This is beyond the pale,” added Mr Biden. “It’s a betrayal of the most sacred duty we bear as a nation, to protect and equip our troops when we send them into harm’s way.”
The intelligence was passed up from the US Special Operations forces based in Afghanistan and led to a restricted high-level White House meeting in late March, people familiar with the situation said.
Russia and the Taliban have denied the existence of the programme.
A National Security Council spokesman said “the veracity of the underlying allegations continue to be evaluated”. The CIA and the Defence and State departments declined to comment when previously contacted by The Independent.