Fati, 23, joined Monaco on loan last summer after Hansi Flick made clear the player had no future in his plans at Camp Nou.
He has since scored 10 goals and contributed one assist across 27 appearances in all competitions under Sébastien Pocognoli and has rediscovered the confidence that injuries and declining opportunities at Barcelona had eroded.
Speaking to the winwin website, his father Bori Fati was direct about the family's wishes. "We still don't know what will happen with his future, because he has a contract until 2028, that's for sure. We don't know, but we want him to stay at Monaco because he's happy there."
Monaco hold a non-mandatory purchase option of €11 million — a figure that represents strong value for a player who had been worth significantly more before his injury troubles.
Whether the Principality club activate it will depend partly on Fati maintaining his current form through the final weeks of the season.
Bori Fati also raised the prospect of a Spain World Cup call-up.
"That's what we hope for. In terms of numbers, if he continues playing in these last three matches with Monaco and plays them all, that's what we expect. And when he's called up again for the Spanish national team, he'll give his all, because statistically, as a striker, he has a higher number of goals. But ultimately, we have to wait and see."
Neither the family nor Flick have yet had any direct conversation about Fati's future at Barcelona.
"It depends on Flick, but they haven't spoken yet," his father confirmed.