The Red Devils swiftly dismissed the package last week, according to The Telegraph's James Ducker.
Real have been actively shopping Colombian international Rodriguez all summer following his return from a loan at Bayern Munich, and the club is understood to value the 28-year-old at £46 million (€50M).
That amount, plus the additional £27.6 million the Spanish giants tabled, would fall well short of United's £160-million appraisal of Pogba.
Pogba, 26, has been heavily linked with an Old Trafford exit following June comments about his desire to seek a "new challenge." Both Real and former club Juventus have been tied to the towering French World Cup winner.
Real's modest advance has confirmed suspicions that the team doesn't have enough financial firepower to sign Pogba, according to Ducker. Zinedine Zidane and Co. have already splashed in excess of £270 million this summer on Eden Hazard, Luka Jovic, Ferland Mendy, Eder Militao, and Rodrygo.