Conway has been expected to depart Ashton Gate this summer amid serious interest in him from a number of clubs.
Scottish champions Celtic have been credited with an interest, while there are a host of Championship clubs who are admirers, including Middlesbrough and Burnley.
Conway, 21, does not want to sign a new contract at Bristol City and is keen to move on.
Now Hull City have put a firm offer of around £5m on the table to convince the Robins to part with Conway.
Whether the proposal will be enough for the Ashton Gate side remains to be seen, but the bid is being considered.
Conway has now moved inside the final 12 months of his Robins deal.He scored ten times in 39 appearances in the Championship for Bristol City over the course of last season, including against Hull City.
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