The Colombian has been deemed surplus to requirements at Real Madrid this season and Zinedine Zidane is looking to offload him.
Pogba has been looking for a move away from Old Trafford, with Real his main suitors.
United have slapped a £150m price tag on the Frenchman, so the offer of Rodriguez plus £27.4m falls well below what they are looking for.
Pogba failed to travel to Cardiff for a friendly against AC Milan, but manager Ole Gunnar Solskjaer put his absence down to injury and an administrative error.
The midfielder did end up lining up against Blackburn Rovers Under-23s on Sunday and Solskjaer says he should be fit to play against Chelsea in the first Premier League game of the season next week.