The French champions, fined £49m for breaching Financial Fair Play guidelines only last month, have already paid Chelsea £50m for David Luiz.
They have also shown interest in Branislav Ivanovic, Oscar and Petr Cech but Hazard is the main prize and they believe Chelsea could sell.
It is understood a Hazard representative met Sheik Tamim bin Hamad al-Thani, PSG’s owner in March. PSG can offer Hazard a five-year deal worth £230,000 a week.
Chelsea manager Jose Mourinho hopes to seal a £35m deal for Atletico Madrid’s Diego Costa before the World Cup starts on June 12 and is also keen on his teammates Tiago, Filipe Luis, Koke and Diego Godin.
Talks to sign Barcelona midfielder Cesc Fabregas are ongoing while Chelsea scouts are closely monitoring Carlos Gruezo, the 19-year-old Stuttgart midfielder.
Chelsea announced the Belgian will wear his favoured No 10 shirt next season, a number that had been left vacant last term.
Hazard told the club’s website: ‘I am very pleased to be able to wear the number 10 for Chelsea.
‘It is my favourite number and is one I am used to wearing for the national team, as well as the one worn by some of my heroes from when I was a kid, like Zinedine Zidane.
‘I hope it brings me a lot of luck in the future and I look forward to the fans seeing me in it next season.’