Bastoni joined Inter Milan in 2017 as a 17-year-old for a staggering £28m, before going back to Atalanta on loan.
After another loan spell, this time at Parma, last season, Bastoni is now beginning to assert himself at Inter.
The 20-year-old has gained the trust of head coach Antonio Conte, who now counts on the youngster as a key player.
Bastoni's emergence as one of the most promising defensive players in Serie A has attracted attention from elsewhere.
Inter are understood to have immediately rejected the bids and are resolute that they won't lose one of their most valuable assets.
Conte is keen to recruit players this month but is less keen to see players leave - especially important figures.
For Barcelona, their new coach Quique Setien is believed to be keeping an eye out for a promising centre-back who's good on the ball.
Setien won his first game in charge of La Blaugrana on Sunday night, beating Granada 1-0.
Manchester City, on the other hand, have a more immediate need of a centre-back.
Having failed to replace the departing Vincent Kompany in the summer, City have relied on Fernandinho playing out of position to fill the gap.
But Fernandinho is in the final months of his contract and could end up leaving in the summer, meaning City will be short once again.
If a deal can't be done this month, Barcelona and Manchester City may well return for Bastoni in the summer.
Bastoni has played 11 games in the league for Inter Milan this season.
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