Paramount is a smaller player than Netflix that is known for brands such as CBS News, Nickelodeon and Mission Impossible.
It started submitting offers a few months ago, eventually prompting Warner Bros, owner of HBO and classics from Looney Tunes to Harry Potter, to formally open a bidding process.
Wall Street analysts have long said they believe a Paramount-Warner Bros combination makes sense, because it would give the company the scale to compete against rivals such as Netflix and Disney.
Paramount was also seen as a strong suitor because the relationship between Trump and the Ellison family, including tech billionaire and Republican megadonor Larry Ellison, was expected to help ease the approval process.
But Warner Bros declared Netflix the winner of the auction on Friday, announcing a deal that valued its studio and streaming networks, including HBO, at about $83bn (£62.3bn), including its debt.
It said the sale would proceed after a planned spin-off of other parts of Warner Brothers' business, including CNN, into an independent company.
Paramount's offer values the entire company at $108.4bn, which it said was a better deal. Trump's son-in-law, Jared Kushner, is among the financial partners Paramount is working with as part of the deal, according to paperwork submitted to the Securities and Exchange Commission.
Netflix executives on Monday expressed confidence in their plans, dismissing Paramount's attempt as "entirely expected".
Warner Bros said it would review the offer but was not currently changing its recommendation. It said it would respond within 10 business days.