The Sun says his move stalled over the fee, with United pointing to a £16million release clause in Jones' contract.
But Rovers owners Venky's claim they are under no obligation to sell at that price and want about £25m.
United are reluctant to improve their £16.5m opening offer but may agree a compromise of about £20m.
The stand-off was still in place last night - but the Jones camp were hopeful the deal can be rubberstamped as early as today.
Old Trafford sources have rejected claims they tapped up Jones, saying they had Rovers' permission for talks.