The highly-rated right-back was targeted by Manchester United last summer only to remain in France.
In September, Fabinho committed his future to Monaco by signing a two-year contract extension, tying him down until 2021.
The 23-year-old has continued to impress in Ligue 1 though, and his father has indicated a move away could still happen.
Along with the Premier League trio, Barcelona have also looked into signing the Brazil international.
Fabinho has attracted interest at Monaco in France
We like Manchester United & Mourinho's work
And while Fabinho and his father, Joao Roberto Tavares, hold United boss Jose Mourinho in high regard, rivals Manchester City are an attractive prospect.
"We have been in talks with Manchester United, Manchester City, Arsenal & approached by Barcelona,” Fabinho’s father told French TV show J+1.
"We like Manchester United & Mourinho's work, but we have a soft spot for Manchester City."
Pep Guardiola is set to overhaul his City squad in the summer, with a right-back likely to be on his agenda.
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Bacary Sagna and Pablo Zabaleta could be without a future at the Etihad, and as both full-backs are ageing, Fabinho would be a long-term solution in defence.
However, Mourinho may yet go back in for the Monaco star to provide competition for Antonio Valencia at right-back.
Arsenal, meanwhile, managed to secure Hector Bellerin’s future at the club in December, suggesting they are not in the running as much.