City are one of four Championship clubs rumoured to be tracking the AFC Wimbledon talisman with the club reluctant to let him go.
The Robins already have Alex Scott and Andreas Weimann for attacking midfielders but an injury to Antoine Semenyo could see the Austrian pushed further up the pitch.
There is also an uncertainty of the future of midfielder Han-Noah Massengo who is yet to sign a new contract with the club with a year remaining.
The current bids of £500,000 are short of the valuation of the player though journalist Richard Cawley believes that at least one of the clubs chasing Rudoni will meet that figure.
Earlier in the month, The Sun claimed that City, Huddersfield and Sunderland had all made bids of £1m that were rejected but that is more than half of what the current offer is.
Despite relegation from League One last season, Rudoni was one of their star performers and got 12 goals and five assists in a campaign that ended their stay in the third tier. Wimbledon have begun their pre-season training and along with Bristol Rovers target Luke McCormick, have returned back on time.
Nigel Pearson admitted last week that he would like three more arrivals in addition to the summer signings of Kane Wilson, Kal Naismith and Mark Sykes.
All three signings this summer have come on a free transfer as the club tries a more astute approach to the transfer market after spending big fees in recent years in the hope of recouping that back.
A small nominal fee for Kasey Palmer from Coventry City came but the 58-year-old said that any deals for any new incomings will be dependent on the structure of the deals.
He said: "The financial aspects of it as well, may dictate it. I don’t want to paint a picture that we’re in financial difficulties. What we have to do is manage our situation with a realistic and sensible touch, and that’s the bottom line.
"I’ll be the blocker, more often than not, than the hierarchy, in terms of whether we do deals or not. Because if deals become too expensive, I’ll just say no, we’re not doing it.
"Because I don’t want to go down that route again, I would like us to have an element of equality amongst the squad but, of course, I never begrudge people earning good money, but we have to have a slightly different approach to what we’ve had in the past, for lots of different reasons."