The Tottenham left-back was wanted by Mourinho in the summer and the United manager is now ready to bring the England international to Old Trafford this month.
But according to the Daily Mirror, Woodward is suggesting cheaper and more inexperienced options in conversations with Mourinho.
The report also claims that the United manager has been told if he wants to sign Rose this month, he will need to offload some of the players in his squad.
Last week, Mourinho said that United’s £300 million outlay on transfers since he arrived at Old Trafford is ‘not enough’.
Mourinho has been unable to keep pace with Manchester City at the top of the Premier League, with Pep Guardiola’s side 15 points clear after their 3-1 win over Watford on Tuesday night.
Rose, meanwhile, is set for another spell on the sidelines after missing Tottenham’s 2-0 win away to Swansea City.
The 27-year-old did not travel to Swansea due to a knee injury and will now undergo tests to find out the extent of the problem.
More: Manchester United FC