Liverpool boss Rafa Benitez intends to push for Heinze once he closes a deal for Atletico Madrid striker Fernando Torres this week.
However, the Liverpool Echo says United have reacted with "horror" to Liverpool's interest and will do all they can to prevent the player moving to Anfield.
United were willing to listen to offers for Heinze, who has slipped behind Patrice Evra in the pecking order at Old Trafford, and it is unclear whether he Sir Alex Ferguson now wants to keep the player or look to offload him overseas.