The New Yorker magazine in an exclusive report sourced the sensational allegations to a second dossier authored by British spy Christopher Steele.
According to the report, Russia opposed Romney because the Republican leader during his 2012 presidential campaign had supported a tough line against the country.
The New Yorker said the second memo, which did not surface publicly with the others, is shorter than the rest, and is based on one source, described as "a senior Russian official".
During Romney's run for president in 2012, he was notably hawkish on Russia, calling it the single greatest threat to the US, the media outlet said.
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"The memo said that the Kremlin, through unspecified channels, had asked Trump to appoint someone who would be prepared to lift Ukraine-related sanctions, and who would cooperate on security issues of interest to Russia, such as the conflict in Syria. If what the source heard was true, then a foreign power was exercising pivotal influence over US foreign policy and an incoming president," it said.
"As fantastical as the memo sounds, subsequent events could be said to support it. In a humiliating public spectacle, Trump dangled the post before Romney until early December, then rejected him," the report said.
"There are plenty of domestic political reasons that Trump may have turned against Romney. Trump loyalists, for instance, noted Romney's public opposition to Trump during the campaign. Roger Stone, the longtime Trump aide, has suggested that Trump was vengefully tormenting Romney, and had never seriously considered him," the magazine said.
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