Rangers made a reported bid of B£350,000 for the Northern Ireland international but their offer has fallen short of Kilmarnock’s valuation of the 23-year-old, who is out of contract in the summer of 2019.
Rangers hope to complete the loan signing of Jason Cummings on Monday and have also opened talks over a potential loan deal for Norwich and Scotland defender Russell Martin.
Cummings arrived in Glasgow on Sunday night ahead of a medical after being left out of the Nottingham Forest squad for Saturday’s defeat by Aston Villa, Aitor Karanka’s first match since replacing Mark Warburton as boss.
The 22-year-old has scored four goals in 10 starts and seven substitute appearances since leaving Hibernian for Forest in the summer.
Martin, 32, has not featured for Norwich since a 4-0 defeat to Millwall on August 26 and subsequently lost his place in the Scotland squad.
Rangers also made an enquiry over Hamilton midfielder Greg Docherty last week.
Graeme Murty integrated new signings Jamie Murphy and Sean Goss in his Light Blues squad during a week-long training camp in Florida, with Andy Halliday and Michael O’Halloran also both featuring during two victories over Brazilian opposition.
The Rangers manager told the club’s website: “Everyone who played has added something to the collective and that is what we want, we want more good players in the squad pushing and improving the people around them and challenging for places.
“If we get that, training will be better, games will be better and hopefully if we react in a positive way we can push ourselves to a higher place.”
Watford are being linked with a summer swoop for Werder Bremen's Sweden international left-back Ludwig Augustinsson.
West Ham are reportedly set to pull out of their plan to sign central defender Alfie Mawson from Swansea this summer.