The 19-year-old right back impressed City manager Roberto Mancini when Leeds met the Sky Blues in the FA Cup, and has since been short listed as one of the English players Mancini would like to sign in the summer.
The package deal would include £6m upfront, as well as add-ons and the possibility of the teenager returning to Elland Road on loan to continue his development.
The energetic youngster has registered interest from West Brom, Stoke and Wigan in recent months, but given the weighty deal Mancini is prepared to offer, City are the favourites to sign the right back.
As the current league champions arrange their deal for Byram, the club faces injury trouble at the Etihad, after David Silva confessed that he fears he may not be fit for City’s FA Cup semi-final match against Chelsea on Sunday.
Silva is scheduled to have scans done today to assess the hamstring strain he sustained during the Manchester Derby on Monday.
Of the injury, which caused him to limp out of the match, the Spanish forward said: “I felt it then and I feel it now.
“It will be a big blow for me. It’s a game that everyone wants to play.”
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