Arsenal had been heavily linked with the French World Cup winner this summer but failed to clinch a deal for the 21-year-old before Thursday’s 5pm Premier League deadline.
And with the European deadline stretching to the end of the month, it is thought more offers are likely to come in for the want-away star, with Paris Saint-Germain the front runners.
The French club have reportedly not only registered interest but have tabled two bids, according to Mundo Deportivo.
The Spanish outlet suggest PSG boss Thomas Tuchel is very keen to reunite with a player he worked with at Borussia Dortmund.
Dembele has previously described Tuchel as one of the best three coaches in the world, alongside ex-Real Madrid boss Zinedine Zidane and Manchester City’s Pep Guardiola.
Barcelona officials are supposedly unsure whether they are willing to let go of a player they only signed a year ago for a staggering £135m.
It is thought that the Catalan side are willing to listen to appropriate offers for their man, but would much prefer to keep him on their books.
Speaking ahead of tonight’s Spanish Super Cup clash, Barca manager Valverde refused to rule out a potential sale but issued a firm warning to the Frenchman.
“I do not know if Dembele can leave. I am the coach and I want him to stay,” he revealed.
“The coach has him and does not think about players coming out.”
Originally from a town 45 miles north-west of the French capital, Dembele has family links in northern France.
A move therefore to the French champions would make logical sense, especially with PSG’s huge spending power demonstrated last summer with the acquisitions of Neymar and Kylian Mbappe for a combined £366m.
However these players, along with the likes of Edinson Cavani and Angel Di Maria, could well be the reason PSG decide not to swoop in for Dembele.
Competition for attacking places is already rife at the Parc des Princes, and unless Tuchel were to offload some of his star forwards, Dembele could struggle to slot in.
PSG have so far only brought in Juventus legend Gianluigi Buffon on a free transfer.
But with a squad already full of star names, Tuchel is unlikely to make further signings unless the right deal were to present itself.
The arrival of Brazilian winger Malcom at from Bordeaux for £36m at the Nou Camp is sure to unnerve Dembele.
There is already competition on the wings from the likes of Lionel Messi, Phillipe Coutinho and Munir.
He had struggled last season to find game time following a hamstring injury early in the season, managing just 17 league appearances.
Barcelona take on Sevilla tonight in the Spanish Super Cup in Morocco.