The 21-year-old has frequently been linked with a move to the Premier League, but but none of England's biggest clubs were able to push through a deal before the start of the new season.
However, the rest of Europe are still able to do business, and according to Mediaset Premium, Roma have made a bid to sign the Jamaica international before the end of play on Friday.
The report claims that the Italian giants have indicated that they are willing to part with €40m (£35.77m), although Leverkusen's valuation remains towards the €50m (£44.71m) mark.
Bailey still has four years remaining on his contract, but it has previously been suggested that the attacker will sign for a high-profile team in Europe sooner rather than later.
Roma have money to spend after missing out on the acquisition of Malcom, who has instead opted to join Barcelona.