Rodriguez appeared to sustain a serious injury to his right knee after landing awkwardly, resulting in him being taken from the field on a stretcher.
The full extent of the injury will not be known until the swelling subsides, but Pochettino fears the worse for the 24-year-old.
"To be completely honest, it is not looking good," the Saints boss said after the 4-1 defeat in Manchester.
"We need to assess him today and tomorrow, but it is not looking good. It is not going to be good for him, not good for the team.
"I can't really know exactly what is wrong with him and when I am going to see him again in a Southampton shirt.
"We need to analyse him today and tomorrow to know the exact reality of his injury.
"All the players are really affected, not just about the result and the refereeing decisions but more than anything the doubt about how severe his injury is."
Asked if he thought it was ligament damage, Pochettino said: "It is possible but at the moment I can't really say anything at all because we have not analysed him yet."
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