The rewards were purportedly incentives to target American forces as Trump tries to withdraw soldiers from the conflict-torn country - one of the militants' key demands - and end America's longest war.
The newspaper said Trump was briefed on the US intelligence findings in March, but has not decided how to respond.
Early on Sunday morning, Trump had criticised the NYT report as "probably just another phony Times hit job, just like their failed Russia Hoax".
"Nobody briefed or told me, @VP Pence, or Chief of Staff @MarkMeadows about the so-called attacks on our troops in Afghanistan by Russians, as reported through an 'anonymous source' by the Fake News @nytimes," he tweeted.
"Everybody is denying it & there have not been many attacks on us... Nobody's been tougher on Russia than the Trump Administration."
John Ratcliffe, the new director of national intelligence, issued a statement late on Saturday denying Trump or Pence had been briefed "on any intelligence alleged by The New York Times in its reporting".
He also vouched for Saturday's White House statement that denied the president had been briefed on the intelligence, but left open the possibility that it existed.
The Taliban have denied the report, reiterating they were committed to a February accord signed with Washington that paves the way for withdrawing all foreign forces from Afghanistan by next year.
The group, widely believed to have received years of support from Pakistani intelligence, also denied previous US accusations it was given arms by Russia.
Russia also denounced the report, with its embassy in Washington tweeting the "baseless and anonymous accusations" in the Times story had "already led to direct threats to the life of employees" at its embassies both there and in London.