The Daily Mail says Betsy is believed to be emerging as the club's first-choice candidate to replace the dismissed John Yems ahead of next season.
Highly-rated Betsy, 44, came through the ranks at the Football Association to manage at various England junior levels up to the Under-18s.
The former Barnsley and Wycombe midfielder is widely recognised as one of the best emerging coaches in the country.
He was also interesting Swindon and Cheltenham, who could be looking for new managers soon.
Crawley finished 12th in the fourth tier last season, 16 points off the play-offs and 23 points clear of the relegation zone.