The Serie A outfit are believed to have made contact with United regarding a potential transfer.
The Dutchman has dropped down the pecking order at Old Trafford, making just three starts in the Premier League under Jose Mourinho this season.
However, Danny Blind has insisted that his son will not be joining Roma in this window.
"Time is running out because it's the last day of the transfer window," the Dutchman told LaRoma24. "Yes, I can confirm that Roma want him but Manchester United don't want to let him go on loan, only on a permanent basis.
"They don't want to sell him now, only in the summer. The price is good, but the operation is difficult. It's really difficult because Manchester United want to keep him to play in the Champions League, the Premier League and the FA Cup."
Roma reportedly wanted to strike an initial loan deal with an option to buy.