The Belgium international has been at Goodison Park since 2013, initially on-loan before signing a permanent deal, and has scored 50 Premier League goals.
PSG reportedly see Lukaku as the best man to fill the void left by Zlatan Ibrahimovic last summer, although Juventus are also set to fight for his services.
The French champions have deployed Edinson Cavani as the central striker this term but it is understood that they are keen to ease the burden on the Uruguay international.
Lukaku is contracted to Everton until 2019 but could be persuaded to depart as Unai Emery will offer Champions League football and the opportunity to fight for the Ligue title.
Ipswich striker Joe Garner will not play again this season after it emerged he suffered a fractured skull against Sheffield United last week.