The Daily Mail claim Jenkins, who is out of contract at the end of the season, had a meeting with Laporte last week and could be the latest Welsh star to cross the Channel.
Toulon owner Mourad Boudjellal has spent a fortune at Stade Felix Mayol and has made the front-row his priortiy for this summer's recruitment.
The Wales star, who has made 11 appearances for the Lions, is keen to settle his future ahead of the RBS 6 Nations which kicks-off next month.
Jenkins would become the latest Welsh Lion to move abroad. 2008 tourists James Hook, Mike Phillips and Lee Byrne already ply their trade in the Top 14 after making summer moves from the Ospreys while former Lions Andy Powell and Dwayne Peel play in England.