But the Colombian - who was desperate to move to the [Black Cats](http://www.thenorthernecho.co.uk/sport/football/sunderland) in the summer transfer window - is yet to respond to Bologna's offer.
Lucumi was [Sunderland's](https://www.thenorthernecho.co.uk/sport/football/sunderland/?ref=ed_direct) leading defensive target throughout the summer but Bologna stood firm and wouldn't allow the 27-year-old to leave, despite the pleas of the centre-back and his agent.
Sunderland turned their attention elsewhere and strengthened their defence with the signings of Omar Alderete, Nordi Mukiele, Arthur Masuaku and Lutsharel Geertruida.
But reports in Italy claim the Black Cats are still interested in Lucumi and are monitoring the defender's situation ahead of the January transfer window.
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If indeed that is the case, any Sunderland move for Lucumi will likely hinge on whether Bologna are successful in trying to tempt the defender to sign a new deal.
Reports in Italy claim the new deal would run until 2029 and would see Lucumi receive a significant salary rise, but it still wouldn't match what he was being offered by Sunderland.
Lucumi's current contract runs until 2027, so if the defender fails to sign a new deal, Bologna face the prospect of seeing the defender move into his last year unless they sell him in January.
Bologna chief executive Claudio Fenucci said this week: “We’re waiting for a response. We’re happy he’s staying, especially since it wasn’t possible to find a replacement for him at the end of the transfer window.
“Given his reaction on the pitch, I’d say he’s behaving like a professional and is dedicating himself entirely to Bologna.”
Bologna have had a disappointing start to the Serie A season, losing two and winning just one of their first three games. Lucumi has featured in all three games.
In the summer, Lucumi's agent Simone Rondanini claimed Bologna had agreed to allow the defender to leave "if the right offer arrived".
But the club's chief executive Fenucci said Lucumi couldn't be sold because Bologna wouldn't be able to replace him.