A meeting has already taken place between Mbappe's representatives and the Catalan giants, with the player telling Monaco that he wants to leave this summer, according to the paper.
Real Madrid already had a £103m bid turned down for the 18-year-old striker earlier this summer, with Monaco holding out for a fee of £178m.
Barcelona will be able to meet Monaco's asking price from the windfall they are in line to receive from the sale of Neymar – the £198m deal with PSG is by far the biggest transfer in football history, dwarfing the £89m Manchester United paid to Juventus for Paul Pogba last summer.
The Catalan giants have earmarked Liverpool midfielder Philippe Coutinho, Borussia Dortmund forward Ousmane Dembele and Juventus forward Paulo Dybala among the other potential replacements for Neymar.
Mbappe was named the Ligue 1 Young Player of the Year after scoring 15 goals in Monaco's title-winning campaign.
The France international said in May that his sole focus was on developing his game by playing at a club that offers regular first-team football.
"I will make a choice based on my sporting career, and my career is first and foremost, and I am at an age where I need to play, and that will be paramount," he told beIN Sports.
"I'm not 35 or 36, an age where I have to manage [my body], I still have everything to learn, this year was discovery, and now comes the most important step: confirmation.
"And I will be the sole decision-maker, and I will not go anywhere where I will be forced to go."