Lavreysen, a five-time Olympic champion, will have several opportunities between now and the end of the championships in Denmark to claim the record for himself.
On Thursday, he races in the keirin, competes in the time trial on Friday and then the individual sprint event on Sunday.
With Jeffrey Hoogland and Roy van den Berg, the Dutch team dominated Australia in Wednesday's final while Japan beat Britain in the bronze medal match.
Lavreysen now has six team sprint titles, five individual and three keirin titles.
Tournant won nine world titles in the team sprint, one in the individual event and four in the time trial between 1997 and 2008.
Lavreysen has built an incredible track record in five years, including a hat-trick (individual and team sprint, keirin) at the Paris Olympics in August, to become the most successful Dutch athlete in the history of the Summer Games.