May’s two-try performance during a 54-29 European Champions Cup drubbing of Castres made it nine touchdowns in eight games since his pre-season move to Leicester Tigers from Gloucester.
Jones, who watched from the Welford Road stands as May and company cut loose, could find it difficult to ignore him for England’s autumn opener against Argentina at Twickenham on November 11.
“I came here to get better and test myself, and I am enjoying the challenge at the moment,” said May. “I physically feel sharper than ever, and I am learning a lot each week. I definitely feel like I am working hard and getting better.
“I am desperate to play for England, but I’ve got to focus on myself in order to do that.”
France forward Yacouba Camara will miss their autumn international series with an ankle injury.
The Montpellier flanker, who has won six caps since his debut against Italy in last year’s Six Nations, faces at least six weeks on the sidelines after suffering the injury in Sunday’s Champions Cup loss to Exeter.
A nasty broken ankle had already sidelined France fly-half Camille Lopez for several months.