The 21-year-old midfielder is out of contract on the finish of the season and attracting consideration from a number of golf equipment.
According to TeamTalk, there may be Premier League pastime from Newcastle, Leicester, Bournemouth and Burnley.
Championship play-off hopefuls Derby also are prepared.
Brownhill is in talks with City over a brand new contract.
He has an choice for an additional 12 months however Robins chiefs are desperate to tie him right down to a longer-term deal.
The Warrington-born ace moved to Ashton Gate upon the expiry of his Preston contract in 2016.
He has made 54 appearances for the membership, showing 22 instances this time period in all competitions this time period.
City are 3rd within the Championship desk and would most likely be reluctant to promote.
Brownhill has prior to now indicated he’s glad on the membership.
When requested in September if he has a long-term long term at Ashton Gate, he mentioned: “Yeah, this is a nice membership and a great fan base and nice supervisor and if the whole thing is in the appropriate position you wish to have to expand as a participant and stay taking part in.
“So I don’t see why now not. I really like the membership, love the fanatics and as I say I imagine what the chief has to provide and the best way he performs soccer.”