The 22-year-old stopper has impressed in his first full season as Forest number one, attracting interest from the likes of Spurs and Liverpool.
The report claims Liverpool's interest is stronger, with Brendan Rodgers thought to be keen to bring in proper competition for first-choice Simon Mignolet.
The £9m Belgian international has made several errors in his first campaign in a Liverpool shirt.
And while Rodgers is reportedly confident Mignolet is his long-term number one, he wants to bring in another talented youngster to push him to perform to his best.
And Darlow has been impressive in England's second tier. The Forest academy graduate was handed the number one shirt at the beginning of the season after a strong finish to last season.
The report claims Liverpool will test Forest's willingness to sell with an initial offer before the deadline, with Spurs waiting in the wings as well.
The north London side are in a slightly different position - Hugo Lloris has been excellent in goal but with Gomes on his way out and Brad Friedel nearing retirement, Tottenham are thought to be looking for a number two to learn from the France international.