Sky sources understand an offer of £225,000 – an improvement of £50,000 from Wednesday – was rejected by the Edinburgh side’s chief executive Leeann Dempster on Thursday afternoon.
Allan was the Player of the Year in the Scottish Championship last season and has been linked with a move to Ibrox for some time.
The 23-year-old grew up supporting Rangers but said last month he was focused on the campaign ahead with Hibs.
Allan, a former Dundee United youth, has been impressive in pre-season, scoring goals in friendly wins against Berwick and Dunfermline Athletic.
He has one year remaining on his current deal and will be free to talk to other clubs about a Bosman move next summer from January 1.
Rangers manager Mark Warburton is keen to add Allan to his expanding squad after bringing in seven first-team signings so far since he was appointed in June.
Rob Kiernan, Danny Wilson, Wes Foderingham, Andy Halliday, Martyn Waghorn, James Tavernier and Jason Holt have all penned contracts.
Rangers travel to play Hibernian in the first round of the Challenge Cup this weekend having met them at the same stage of the same competition last term.
Warburton is keen to have Allan in his pool in time for that game but Hibs are standing firm for now and refusing to let the player leave.