Ayew has received plenty of plaudits for a collection of fine performances for Crystal Palace in the Premier League this season.
The 28-year-old currently stands as The Eagles' top scorer for the campaign, having notched nine goals in 30 appearances so far.
"It’s time for him [to step up]," Desailly, who spent six years at Chelsea between 1998 and 2004, told FootballmadeinGhana.
"He’s 28-years-old now and he has reached a very high level, as well as being quite consistent in his performances.
“We need him to have more hunger. We want him to become a killer.
"You can see in his face that he is not really a killer, but he is showing great ability, and being at Chelsea can bring him to the level that we all know that he can get to because he is a very good player."
Ayew's last goal - in a 2-0 away triumph over Bournemouth - did not just draw him close to double figures for the season but also put him clear as the highest-scoring Ghanaian in the history of the Premier League with 25 goals.
He sits one goal above former Leeds United forward Anthony Yeboah.
He reached the 25-goal mark in 129 matches while Yeboah scored his 24 goals in 47 appearances.
Ayew is also on the brink of breaking another record, this time at Palace, with seven more matches to the end of the season.
The attacker's goals so far have won Palace 14 points this term, leaving him just one point behind Andrew Johnson’s contribution in 2004-05 as no single player has collected The Eagles more points in Premier League history.
Ayew joined Palace in January 2019 for a half-season loan stint from Swansea. Despite a frustrating spell which yielded only one league goal, the striker was signed on a permanent deal at the end of the season.
He first came to England in 2015, signing for Villa from French side Lorient. In January 2017, he left for Swansea.
While in France, Ayew also played for Sochaux and Olympique Marseille.