The Foxes sent a scout to Lisbon on Tuesday for the Champions League clash between Benfica and Basel, which the Swiss side won 2-0, with goalscorer Elyounoussi believed to be the target.
The 23-year-old Norwegian, who plays on the left of a three-man forward line for the Swiss champions, has been on the club's radar for the past year and was linked with a move to the King Power Stadium in the summer.
Southampton were also reportedly keen at the time, which was one month before Puel was fired from the St Mary's hotseat.
And the Saints are continuing to monitor the ex-Molde player with Swiss newspaper Blick claiming that the south coast club sent a spy to watch Elyounoussi play against Lausanne on Saturday night.
They report that the Southampton scout was "zealously recording his observations whenever Elyounoussi was on the ball" and that "at the end of the game, his notebook was full".
Elyounoussi, who is under contract until 2021, is expected to fetch several million euros when he does leave Basel.