In emails to Trump lawyer Michael Cohen that the newspaper cited, the associate, Felix Sater, wrote: "Our boy can become president of the USA and we can engineer it. I will get all of Putins team to buy in on this, I will manage this process."
In a separate email seen by the Times, Sater, a Russian immigrant and then-Trump Organization broker, wrote: "I will get Putin on this program and we will get Donald elected."
The development comes as federal and congressional investigators probe Russian efforts to influence the 2016 election and whether the Trump campaign coordinated with the Kremlin. Trump has denied any collusion, though developments since the election have shown that Trump associates were at least willing to accept Russian help during the election.
Cohen told the Times that Sater practices "salesmanship" and that he "ultimately determined that the proposal was not feasible." Cohen "never agreed" to make a trip to Russia, he added.
It is not clear why Sater would have thought a Trump Tower project in Moscow would have helped the president's chances of winning.
Separately on Monday, The Washington Post reported that Cohen asked top Kremlin press aide Dmitry Peskov for help with the "stalled" Trump Tower project in January 2016.
"Over the past few months I have been working with a company based in Russia regarding the development of a Trump Tower – Moscow project in Moscow City," Cohen wrote Peskov, the Post reported, citing a person familiar with the email. "Without getting into lengthy specifics. the communication between our two sides has stalled."
"As this project is too important, I am hereby requesting your assistance. I respectfully request someone, preferably you, contact me so that I might discuss the specifics as well as arranging meetings with the appropriate individuals. I thank you in advance for your assistance and look forward to hearing from you soon," he also wrote, according to the Post.
Cohen told congressional investigators that Sater recommended that he write the email, the Post added. He added that "he did not recall receiving a response," the newspaper reported.
A plan to build a Trump Tower in Moscow never came about. In a statement to the Times on Monday, the Trump Organization said it "has never had any real estate holdings or interests in Russia."