Argentina defender Rojo is the subject of disciplinary action by the Portuguese club after he handed in a transfer request and refused to train amid reports of a bid from United.
Sky sources in Lisbon reported that United bid around £16million but it is believed Sporting are holding out for more as they only own 25 per cent of the player.
Management company Doyen Sports, which owns 75 per cent of the player, has hit back at Sporting's stance and threatened legal action if fees due from any rejected bid are not paid.
It is reported that Doyen is entitled to a payment from Sporting worth 75 per cent of any declined offer for Rojo. In this instance that would equate to £12million.
In a statement, Doyen Sports has stressed it cannot interfere in transfer business and that Sporting have a right to retain the player who has a £24million release clause.
But it has pointed out that in such a scenario the club still have obligations to the company.
The statement read: "Sporting is entirely within its rights not to transfer the player Marcos Rojo, knowing that it only has to make up for the fund under the terms and deadlines as contractually established since the beginning."
It added the company "will not hesitate to use all legal resources at our disposal to defend fully all our interests and rights".
Doyen holds strong rights over the player after paying 75 per cent of his transfer fee from Spartak Moscow in 2012.
The statement added: "Without the intervention of Doyen, through financing, Marcos Rojo would not be a Sporting player."
Sporting have responded with their own detailed statement which alleges interference from Doyen.
They insist they have had "just cause" to terminate their agreement with the company, therefore suggesting they do not intend to pay any fees.
Sporting have also confirmed that Spartak would be owed 20 per cent of any fee received for Rojo above £4million.