Derek McInnes' side, whose side sit second bottom of the npower Championship, are understood to have made a bid for the 22-year-old, who made his Scotland debut as a half-time substitute in a 2-1 friendly win over Luxembourg in November.
The former Milton Keynes Dons player is believed to be keen to return to England to be closer to his mother following the death of his father, who collapsed with a heart attack at Hampden as Kilmarnock won the Scottish Communities League Cup final against Celtic in March.
But Kelly has not rejected the offer of a new contract from Kilmarnock, where his deal expires at the end of the season.
City are reported to have offered £200,000 for Kelly, who has scored six goals in his last nine games for Kilmarnock.
But Killie boss Kenny Shiels told STV: "Liam is happy to stay with us. I'm sure he'll get plenty of offers but hopefully we can keep him at the club.
"In fact we would be keen to tie him to a new deal."